The Alternating Knot 10₁₀₈

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Acknowledgement

KnotPlot

PD Presentation: X₆₂₇₁ X_16,4,17,3 X_20,13,1,14 X_14,7,15,8 X_8,19,9,20 X_18,9,19,10 X_10,17,11,18 X_12,6,13,5 X_4,12,5,11 X_2,16,3,15

Gauss Code: {1, -10, 2, -9, 8, -1, 4, -5, 6, -7, 9, -8, 3, -4, 10, -2, 7, -6, 5, -3}

DT (Dowker-Thistlethwaite) Code: 6 16 12 14 18 4 20 2 10 8

Minimum Braid Representative:

Length is 11, width is 4
Braid index is 4

A Morse Link Presentation:

3D Invariants:

Symmetry Type Unknotting Number 3-Genus Bridge/Super Bridge Index Nakanishi Index

Reversible 2 3 3 / NotAvailable 1

Alexander Polynomial: 2t^-3 - 8t^-2 + 14t^-1 - 15 + 14t - 8t² + 2t³

Conway Polynomial: 1 + 4z⁴ + 2z⁶

Other knots with the same Alexander/Conway Polynomial: {K11n161, ...}

Determinant and Signature: {63, 2}

Jones Polynomial: - q^-4 + 3q^-3 - 5q^-2 + 8q^-1 - 9 + 10q - 10q² + 8q³ - 5q⁴ + 3q⁵ - q⁶

Other knots (up to mirrors) with the same Jones Polynomial: {...}

A2 (sl(3)) Invariant: - q^-12 + q^-10 + 2q^-4 - q^-2 + 2 - q⁴ + q⁶ - 2q⁸ + 2q¹⁰ + q¹⁶ - q¹⁸

HOMFLY-PT Polynomial: - 2a^-4z² - a^-4z⁴ + 2a^-2z² + 3a^-2z⁴ + a^-2z⁶ + 1 + 2z² + 3z⁴ + z⁶ - 2a²z² - a²z⁴

Kauffman Polynomial: a^-7z³ - a^-6z² + 3a^-6z⁴ - 3a^-5z³ + 5a^-5z⁵ + 2a^-4z² - 9a^-4z⁴ + 7a^-4z⁶ - 2a^-3z + 10a^-3z³ - 17a^-3z⁵ + 8a^-3z⁷ + 4a^-2z⁴ - 13a^-2z⁶ + 6a^-2z⁸ - 6a^-1z + 28a^-1z³ - 29a^-1z⁵ + 4a^-1z⁷ + 2a^-1z⁹ + 1 - 10z² + 33z⁴ - 33z⁶ + 9z⁸ - 6az + 19az³ - 11az⁵ - 3az⁷ + 2az⁹ - 7a²z² + 17a²z⁴ - 13a²z⁶ + 3a²z⁸ - 2a³z + 5a³z³ - 4a³z⁵ + a³z⁷

V₂ and V₃, the type 2 and 3 Vassiliev invariants: {0, 0}

Khovanov Homology:
(The squares with yellow highlighting are those on the "critical diagonals", where j-2r=s+1 or j-2r=s+1, where s=2 is the signature of 10₁₀₈. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

t^rq^j r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5

j = 13 1

j = 11 2

j = 9 3 1

j = 7 5 2

j = 5 5 3

j = 3 5 5

j = 1 5 6

j = -1 3 4

j = -3 2 5

j = -5 1 3

j = -7 2

j = -9 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-13 - 3q^-12 - q^-11 + 11q^-10 - 9q^-9 - 14q^-8 + 30q^-7 - 5q^-6 - 39q^-5 + 43q^-4 + 11q^-3 - 62q^-2 + 42q^-1 + 32 - 73q + 30q² + 47q³ - 67q⁴ + 13q⁵ + 47q⁶ - 47q⁷ + 2q⁸ + 30q⁹ - 25q¹⁰ + 3q¹¹ + 11q¹² - 11q¹³ + 4q¹⁴ + 2q¹⁵ - 3q¹⁶ + q¹⁷

3 - q^-27 + 3q^-26 + q^-25 - 5q^-24 - 9q^-23 + 9q^-22 + 23q^-21 - 6q^-20 - 42q^-19 - 12q^-18 + 61q^-17 + 43q^-16 - 66q^-15 - 84q^-14 + 53q^-13 + 121q^-12 - 16q^-11 - 151q^-10 - 25q^-9 + 150q^-8 + 82q^-7 - 143q^-6 - 122q^-5 + 109q^-4 + 164q^-3 - 77q^-2 - 183q^-1 + 26 + 208q + 17q² - 213q³ - 70q⁴ + 215q⁵ + 118q⁶ - 201q⁷ - 162q⁸ + 172q⁹ + 191q¹⁰ - 132q¹¹ - 195q¹² + 85q¹³ + 173q¹⁴ - 38q¹⁵ - 138q¹⁶ + 13q¹⁷ + 87q¹⁸ + 5q¹⁹ - 49q²⁰ - 4q²¹ + 17q²² + 5q²³ - 6q²⁴ + q²⁵ - 3q²⁶ + 4q²⁸ - q²⁹ - q³⁰ - 2q³¹ + 3q³² - q³³

4 q^-46 - 3q^-45 - q^-44 + 5q^-43 + 3q^-42 + 9q^-41 - 19q^-40 - 21q^-39 + 4q^-38 + 18q^-37 + 73q^-36 - 14q^-35 - 74q^-34 - 74q^-33 - 42q^-32 + 187q^-31 + 115q^-30 - 7q^-29 - 172q^-28 - 300q^-27 + 124q^-26 + 236q^-25 + 292q^-24 + 6q^-23 - 524q^-22 - 198q^-21 - q^-20 + 502q^-19 + 483q^-18 - 332q^-17 - 387q^-16 - 554q^-15 + 242q^-14 + 809q^-13 + 194q^-12 - 95q^-11 - 956q^-10 - 359q^-9 + 661q^-8 + 618q^-7 + 529q^-6 - 939q^-5 - 909q^-4 + 187q^-3 + 742q^-2 + 1137q^-1 - 652 - 1247q - 333q² + 693q³ + 1618q⁴ - 298q⁵ - 1481q⁶ - 834q⁷ + 598q⁸ + 2033q⁹ + 126q¹⁰ - 1617q¹¹ - 1355q¹² + 340q¹³ + 2278q¹⁴ + 680q¹⁵ - 1424q¹⁶ - 1718q¹⁷ - 173q¹⁸ + 2043q¹⁹ + 1118q²⁰ - 804q²¹ - 1564q²² - 646q²³ + 1304q²⁴ + 1062q²⁵ - 157q²⁶ - 932q²⁷ - 686q²⁸ + 548q²⁹ + 600q³⁰ + 107q³¹ - 329q³² - 406q³³ + 166q³⁴ + 196q³⁵ + 77q³⁶ - 55q³⁷ - 159q³⁸ + 58q³⁹ + 30q⁴⁰ + 20q⁴¹ + 6q⁴² - 52q⁴³ + 26q⁴⁴ - q⁴⁵ + 3q⁴⁶ + 6q⁴⁷ - 15q⁴⁸ + 8q⁴⁹ - 2q⁵⁰ + q⁵¹ + 2q⁵² - 3q⁵³ + q⁵⁴

5 - q^-70 + 3q^-69 + q^-68 - 5q^-67 - 3q^-66 - 3q^-65 + q^-64 + 17q^-63 + 24q^-62 - 6q^-61 - 31q^-60 - 48q^-59 - 40q^-58 + 26q^-57 + 112q^-56 + 122q^-55 + 24q^-54 - 122q^-53 - 241q^-52 - 203q^-51 + 42q^-50 + 337q^-49 + 433q^-48 + 212q^-47 - 233q^-46 - 641q^-45 - 634q^-44 - 112q^-43 + 611q^-42 + 1013q^-41 + 727q^-40 - 177q^-39 - 1133q^-38 - 1384q^-37 - 636q^-36 + 727q^-35 + 1758q^-34 + 1622q^-33 + 240q^-32 - 1523q^-31 - 2404q^-30 - 1580q^-29 + 587q^-28 + 2568q^-27 + 2843q^-26 + 1000q^-25 - 1900q^-24 - 3680q^-23 - 2757q^-22 + 437q^-21 + 3624q^-20 + 4338q^-19 + 1614q^-18 - 2755q^-17 - 5278q^-16 - 3732q^-15 + 1027q^-14 + 5432q^-13 + 5663q^-12 + 1077q^-11 - 4775q^-10 - 7002q^-9 - 3373q^-8 + 3519q^-7 + 7779q^-6 + 5414q^-5 - 1886q^-4 - 7951q^-3 - 7191q^-2 + 201q^-1 + 7784 + 8515q + 1392q² - 7381q³ - 9620q⁴ - 2763q⁵ + 7043q⁶ + 10498q⁷ + 3968q⁸ - 6748q⁹ - 11423q¹⁰ - 5111q¹¹ + 6554q¹² + 12398q¹³ + 6350q¹⁴ - 6263q¹⁵ - 13395q¹⁶ - 7799q¹⁷ + 5636q¹⁸ + 14225q¹⁹ + 9456q²⁰ - 4520q²¹ - 14553q²² - 11063q²³ + 2765q²⁴ + 14075q²⁵ + 12362q²⁶ - 612q²⁷ - 12675q²⁸ - 12858q²⁹ - 1604q³⁰ + 10374q³¹ + 12407q³² + 3439q³³ - 7654q³⁴ - 10951q³⁵ - 4481q³⁶ + 4912q³⁷ + 8806q³⁸ + 4693q³⁹ - 2673q⁴⁰ - 6440q⁴¹ - 4137q⁴² + 1094q⁴³ + 4288q⁴⁴ + 3200q⁴⁵ - 246q⁴⁶ - 2577q⁴⁷ - 2183q⁴⁸ - 140q⁴⁹ + 1442q⁵⁰ + 1349q⁵¹ + 169q⁵² - 724q⁵³ - 729q⁵⁴ - 152q⁵⁵ + 350q⁵⁶ + 378q⁵⁷ + 71q⁵⁸ - 168q⁵⁹ - 158q⁶⁰ - 26q⁶¹ + 65q⁶² + 65q⁶³ + 13q⁶⁴ - 37q⁶⁵ - 28q⁶⁶ + 15q⁶⁷ + 12q⁶⁸ - 3q⁶⁹ + 5q⁷⁰ - 9q⁷¹ - 6q⁷² + 9q⁷³ + 3q⁷⁴ - 5q⁷⁵ + 2q⁷⁶ - q⁷⁷ - 2q⁷⁸ + 3q⁷⁹ - q⁸⁰

6 q^-99 - 3q^-98 - q^-97 + 5q^-96 + 3q^-95 + 3q^-94 - 7q^-93 + q^-92 - 20q^-91 - 22q^-90 + 18q^-89 + 32q^-88 + 54q^-87 + 17q^-86 + 24q^-85 - 86q^-84 - 160q^-83 - 102q^-82 - 15q^-81 + 166q^-80 + 225q^-79 + 389q^-78 + 110q^-77 - 256q^-76 - 520q^-75 - 642q^-74 - 356q^-73 + 22q^-72 + 997q^-71 + 1172q^-70 + 900q^-69 + 74q^-68 - 1032q^-67 - 1818q^-66 - 2135q^-65 - 477q^-64 + 1015q^-63 + 2648q^-62 + 3043q^-61 + 2026q^-60 - 444q^-59 - 3701q^-58 - 4309q^-57 - 3828q^-56 - 539q^-55 + 3254q^-54 + 6486q^-53 + 6356q^-52 + 2047q^-51 - 2526q^-50 - 7892q^-49 - 8909q^-48 - 5920q^-47 + 1870q^-46 + 9093q^-45 + 11599q^-44 + 9795q^-43 + 653q^-42 - 9093q^-41 - 16202q^-40 - 13447q^-39 - 3753q^-38 + 8574q^-37 + 19433q^-36 + 18881q^-35 + 8435q^-34 - 9668q^-33 - 21983q^-32 - 24182q^-31 - 13154q^-30 + 8683q^-29 + 26259q^-28 + 30926q^-27 + 15632q^-26 - 7793q^-25 - 30257q^-24 - 37102q^-23 - 20087q^-22 + 9674q^-21 + 36660q^-20 + 40860q^-19 + 23013q^-18 - 12262q^-17 - 43104q^-16 - 47116q^-15 - 21990q^-14 + 19289q^-13 + 48281q^-12 + 51100q^-11 + 19166q^-10 - 27811q^-9 - 57425q^-8 - 50697q^-7 - 9827q^-6 + 36874q^-5 + 64244q^-4 + 47771q^-3 - 2564q^-2 - 51600q^-1 - 66601 - 36353q + 17438q² + 64028q³ + 65590q⁴ + 20066q⁵ - 39619q⁶ - 71787q⁷ - 53786q⁸ + 948q⁹ + 59566q¹⁰ + 75008q¹¹ + 34853q¹² - 30977q¹³ - 74653q¹⁴ - 64998q¹⁵ - 8820q¹⁶ + 58906q¹⁷ + 84006q¹⁸ + 45793q¹⁹ - 27420q²⁰ - 81755q²¹ - 78153q²² - 17903q²³ + 61125q²⁴ + 97531q²⁵ + 61840q²⁶ - 20008q²⁷ - 88864q²⁸ - 96530q²⁹ - 36325q³⁰ + 54096q³¹ + 107686q³² + 84205q³³ + 1832q³⁴ - 80956q³⁵ - 108472q³⁶ - 62409q³⁷ + 27996q³⁸ + 97537q³⁹ + 97465q⁴⁰ + 32363q⁴¹ - 50565q⁴² - 96492q⁴³ - 77209q⁴⁴ - 6489q⁴⁵ + 63530q⁴⁶ + 84951q⁴⁷ + 49816q⁴⁸ - 13345q⁴⁹ - 61777q⁵⁰ - 66239q⁵¹ - 26293q⁵² + 25440q⁵³ + 52636q⁵⁴ + 42949q⁵⁵ + 8130q⁵⁶ - 26368q⁵⁷ - 39357q⁵⁸ - 24102q⁵⁹ + 3544q⁶⁰ + 22592q⁶¹ + 23970q⁶² + 10371q⁶³ - 6740q⁶⁴ - 16513q⁶⁵ - 12759q⁶⁶ - 1874q⁶⁷ + 6757q⁶⁸ + 9172q⁶⁹ + 5346q⁷⁰ - 720q⁷¹ - 5207q⁷² - 4487q⁷³ - 1118q⁷⁴ + 1501q⁷⁵ + 2547q⁷⁶ + 1687q⁷⁷ + 122q⁷⁸ - 1401q⁷⁹ - 1128q⁸⁰ - 209q⁸¹ + 269q⁸² + 542q⁸³ + 356q⁸⁴ + 86q⁸⁵ - 366q⁸⁶ - 203q⁸⁷ + 30q⁸⁸ + 27q⁸⁹ + 87q⁹⁰ + 39q⁹¹ + 41q⁹² - 95q⁹³ - 14q⁹⁴ + 33q⁹⁵ - 12q⁹⁶ + 10q⁹⁷ - 9q⁹⁸ + 19q⁹⁹ - 22q¹⁰⁰ + 5q¹⁰¹ + 12q¹⁰² - 9q¹⁰³ + 3q¹⁰⁴ - 6q¹⁰⁵ + 5q¹⁰⁶ - 2q¹⁰⁷ + q¹⁰⁸ + 2q¹⁰⁹ - 3q¹¹⁰ + q¹¹¹

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=

<< KnotTheory`

Loading KnotTheory` (version of August 30, 2005, 10:15:35)...

In[2]:=
PD[Knot[10, 108]]

Out[2]=
PD[X[6, 2, 7, 1], X[16, 4, 17, 3], X[20, 13, 1, 14], X[14, 7, 15, 8], > X[8, 19, 9, 20], X[18, 9, 19, 10], X[10, 17, 11, 18], X[12, 6, 13, 5], > X[4, 12, 5, 11], X[2, 16, 3, 15]]

In[3]:=
GaussCode[Knot[10, 108]]

Out[3]=
GaussCode[1, -10, 2, -9, 8, -1, 4, -5, 6, -7, 9, -8, 3, -4, 10, -2, 7, -6, 5, > -3]

In[4]:=
DTCode[Knot[10, 108]]

Out[4]=
DTCode[6, 16, 12, 14, 18, 4, 20, 2, 10, 8]

In[5]:=
br = BR[Knot[10, 108]]

Out[5]=
BR[4, {1, 1, -2, 1, 1, 3, -2, 1, -2, -3, -3}]

In[6]:=
{First[br], Crossings[br]}

Out[6]=
{4, 11}

In[7]:=
BraidIndex[Knot[10, 108]]

Out[7]=
4

In[8]:=
Show[DrawMorseLink[Knot[10, 108]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[10, 108]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

Out[9]=
{Reversible, 2, 3, 3, NotAvailable, 1}

In[10]:=
alex = Alexander[Knot[10, 108]][t]

Out[10]=
2 8 14 2 3 -15 + -- - -- + -- + 14 t - 8 t + 2 t 3 2 t t t

In[11]:=
Conway[Knot[10, 108]][z]

Out[11]=
4 6 1 + 4 z + 2 z

In[12]:=
Select[AllKnots[], (alex === Alexander[#][t])&]

Out[12]=
{Knot[10, 108], Knot[11, NonAlternating, 161]}

In[13]:=
{KnotDet[Knot[10, 108]], KnotSignature[Knot[10, 108]]}

Out[13]=
{63, 2}

In[14]:=
Jones[Knot[10, 108]][q]

Out[14]=
-4 3 5 8 2 3 4 5 6 -9 - q + -- - -- + - + 10 q - 10 q + 8 q - 5 q + 3 q - q 3 2 q q q

In[15]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]

Out[15]=
{Knot[10, 108]}

In[16]:=
A2Invariant[Knot[10, 108]][q]

Out[16]=
-12 -10 2 -2 4 6 8 10 16 18 2 - q + q + -- - q - q + q - 2 q + 2 q + q - q 4 q

In[17]:=
HOMFLYPT[Knot[10, 108]][a, z]

Out[17]=
2 2 4 4 6 2 2 z 2 z 2 2 4 z 3 z 2 4 6 z 1 + 2 z - ---- + ---- - 2 a z + 3 z - -- + ---- - a z + z + -- 4 2 4 2 2 a a a a a

In[18]:=
Kauffman[Knot[10, 108]][a, z]

Out[18]=
2 2 3 3 2 z 6 z 3 2 z 2 z 2 2 z 3 z 1 - --- - --- - 6 a z - 2 a z - 10 z - -- + ---- - 7 a z + -- - ---- + 3 a 6 4 7 5 a a a a a 3 3 4 4 4 10 z 28 z 3 3 3 4 3 z 9 z 4 z 2 4 > ----- + ----- + 19 a z + 5 a z + 33 z + ---- - ---- + ---- + 17 a z + 3 a 6 4 2 a a a a 5 5 5 6 6 5 z 17 z 29 z 5 3 5 6 7 z 13 z > ---- - ----- - ----- - 11 a z - 4 a z - 33 z + ---- - ----- - 5 3 a 4 2 a a a a 7 7 8 9 2 6 8 z 4 z 7 3 7 8 6 z 2 8 2 z > 13 a z + ---- + ---- - 3 a z + a z + 9 z + ---- + 3 a z + ---- + 3 a 2 a a a 9 > 2 a z

In[19]:=
{Vassiliev[2][Knot[10, 108]], Vassiliev[3][Knot[10, 108]]}

Out[19]=
{0, 0}

In[20]:=
Kh[Knot[10, 108]][q, t]

Out[20]=
3 1 2 1 3 2 5 3 4 5 q 6 q + 5 q + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + --- + 9 5 7 4 5 4 5 3 3 3 3 2 2 q t t q t q t q t q t q t q t q t 3 5 5 2 7 2 7 3 9 3 9 4 > 5 q t + 5 q t + 3 q t + 5 q t + 2 q t + 3 q t + q t + 11 4 13 5 > 2 q t + q t

In[21]:=
ColouredJones[Knot[10, 108], 2][q]

Out[21]=
-13 3 -11 11 9 14 30 5 39 43 11 62 42 32 + q - --- - q + --- - -- - -- + -- - -- - -- + -- + -- - -- + -- - 12 10 9 8 7 6 5 4 3 2 q q q q q q q q q q q 2 3 4 5 6 7 8 9 > 73 q + 30 q + 47 q - 67 q + 13 q + 47 q - 47 q + 2 q + 30 q - 10 11 12 13 14 15 16 17 > 25 q + 3 q + 11 q - 11 q + 4 q + 2 q - 3 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10₁₀₈

10₁₀₇
10₁₀₉

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-13 - 3q^-12 - q^-11 + 11q^-10 - 9q^-9 - 14q^-8 + 30q^-7 - 5q^-6 - 39q^-5 + 43q^-4 + 11q^-3 - 62q^-2 + 42q^-1 + 32 - 73q + 30q² + 47q³ - 67q⁴ + 13q⁵ + 47q⁶ - 47q⁷ + 2q⁸ + 30q⁹ - 25q¹⁰ + 3q¹¹ + 11q¹² - 11q¹³ + 4q¹⁴ + 2q¹⁵ - 3q¹⁶ + q¹⁷
3	- q^-27 + 3q^-26 + q^-25 - 5q^-24 - 9q^-23 + 9q^-22 + 23q^-21 - 6q^-20 - 42q^-19 - 12q^-18 + 61q^-17 + 43q^-16 - 66q^-15 - 84q^-14 + 53q^-13 + 121q^-12 - 16q^-11 - 151q^-10 - 25q^-9 + 150q^-8 + 82q^-7 - 143q^-6 - 122q^-5 + 109q^-4 + 164q^-3 - 77q^-2 - 183q^-1 + 26 + 208q + 17q² - 213q³ - 70q⁴ + 215q⁵ + 118q⁶ - 201q⁷ - 162q⁸ + 172q⁹ + 191q¹⁰ - 132q¹¹ - 195q¹² + 85q¹³ + 173q¹⁴ - 38q¹⁵ - 138q¹⁶ + 13q¹⁷ + 87q¹⁸ + 5q¹⁹ - 49q²⁰ - 4q²¹ + 17q²² + 5q²³ - 6q²⁴ + q²⁵ - 3q²⁶ + 4q²⁸ - q²⁹ - q³⁰ - 2q³¹ + 3q³² - q³³
4	q^-46 - 3q^-45 - q^-44 + 5q^-43 + 3q^-42 + 9q^-41 - 19q^-40 - 21q^-39 + 4q^-38 + 18q^-37 + 73q^-36 - 14q^-35 - 74q^-34 - 74q^-33 - 42q^-32 + 187q^-31 + 115q^-30 - 7q^-29 - 172q^-28 - 300q^-27 + 124q^-26 + 236q^-25 + 292q^-24 + 6q^-23 - 524q^-22 - 198q^-21 - q^-20 + 502q^-19 + 483q^-18 - 332q^-17 - 387q^-16 - 554q^-15 + 242q^-14 + 809q^-13 + 194q^-12 - 95q^-11 - 956q^-10 - 359q^-9 + 661q^-8 + 618q^-7 + 529q^-6 - 939q^-5 - 909q^-4 + 187q^-3 + 742q^-2 + 1137q^-1 - 652 - 1247q - 333q² + 693q³ + 1618q⁴ - 298q⁵ - 1481q⁶ - 834q⁷ + 598q⁸ + 2033q⁹ + 126q¹⁰ - 1617q¹¹ - 1355q¹² + 340q¹³ + 2278q¹⁴ + 680q¹⁵ - 1424q¹⁶ - 1718q¹⁷ - 173q¹⁸ + 2043q¹⁹ + 1118q²⁰ - 804q²¹ - 1564q²² - 646q²³ + 1304q²⁴ + 1062q²⁵ - 157q²⁶ - 932q²⁷ - 686q²⁸ + 548q²⁹ + 600q³⁰ + 107q³¹ - 329q³² - 406q³³ + 166q³⁴ + 196q³⁵ + 77q³⁶ - 55q³⁷ - 159q³⁸ + 58q³⁹ + 30q⁴⁰ + 20q⁴¹ + 6q⁴² - 52q⁴³ + 26q⁴⁴ - q⁴⁵ + 3q⁴⁶ + 6q⁴⁷ - 15q⁴⁸ + 8q⁴⁹ - 2q⁵⁰ + q⁵¹ + 2q⁵² - 3q⁵³ + q⁵⁴
5	- q^-70 + 3q^-69 + q^-68 - 5q^-67 - 3q^-66 - 3q^-65 + q^-64 + 17q^-63 + 24q^-62 - 6q^-61 - 31q^-60 - 48q^-59 - 40q^-58 + 26q^-57 + 112q^-56 + 122q^-55 + 24q^-54 - 122q^-53 - 241q^-52 - 203q^-51 + 42q^-50 + 337q^-49 + 433q^-48 + 212q^-47 - 233q^-46 - 641q^-45 - 634q^-44 - 112q^-43 + 611q^-42 + 1013q^-41 + 727q^-40 - 177q^-39 - 1133q^-38 - 1384q^-37 - 636q^-36 + 727q^-35 + 1758q^-34 + 1622q^-33 + 240q^-32 - 1523q^-31 - 2404q^-30 - 1580q^-29 + 587q^-28 + 2568q^-27 + 2843q^-26 + 1000q^-25 - 1900q^-24 - 3680q^-23 - 2757q^-22 + 437q^-21 + 3624q^-20 + 4338q^-19 + 1614q^-18 - 2755q^-17 - 5278q^-16 - 3732q^-15 + 1027q^-14 + 5432q^-13 + 5663q^-12 + 1077q^-11 - 4775q^-10 - 7002q^-9 - 3373q^-8 + 3519q^-7 + 7779q^-6 + 5414q^-5 - 1886q^-4 - 7951q^-3 - 7191q^-2 + 201q^-1 + 7784 + 8515q + 1392q² - 7381q³ - 9620q⁴ - 2763q⁵ + 7043q⁶ + 10498q⁷ + 3968q⁸ - 6748q⁹ - 11423q¹⁰ - 5111q¹¹ + 6554q¹² + 12398q¹³ + 6350q¹⁴ - 6263q¹⁵ - 13395q¹⁶ - 7799q¹⁷ + 5636q¹⁸ + 14225q¹⁹ + 9456q²⁰ - 4520q²¹ - 14553q²² - 11063q²³ + 2765q²⁴ + 14075q²⁵ + 12362q²⁶ - 612q²⁷ - 12675q²⁸ - 12858q²⁹ - 1604q³⁰ + 10374q³¹ + 12407q³² + 3439q³³ - 7654q³⁴ - 10951q³⁵ - 4481q³⁶ + 4912q³⁷ + 8806q³⁸ + 4693q³⁹ - 2673q⁴⁰ - 6440q⁴¹ - 4137q⁴² + 1094q⁴³ + 4288q⁴⁴ + 3200q⁴⁵ - 246q⁴⁶ - 2577q⁴⁷ - 2183q⁴⁸ - 140q⁴⁹ + 1442q⁵⁰ + 1349q⁵¹ + 169q⁵² - 724q⁵³ - 729q⁵⁴ - 152q⁵⁵ + 350q⁵⁶ + 378q⁵⁷ + 71q⁵⁸ - 168q⁵⁹ - 158q⁶⁰ - 26q⁶¹ + 65q⁶² + 65q⁶³ + 13q⁶⁴ - 37q⁶⁵ - 28q⁶⁶ + 15q⁶⁷ + 12q⁶⁸ - 3q⁶⁹ + 5q⁷⁰ - 9q⁷¹ - 6q⁷² + 9q⁷³ + 3q⁷⁴ - 5q⁷⁵ + 2q⁷⁶ - q⁷⁷ - 2q⁷⁸ + 3q⁷⁹ - q⁸⁰
6	q^-99 - 3q^-98 - q^-97 + 5q^-96 + 3q^-95 + 3q^-94 - 7q^-93 + q^-92 - 20q^-91 - 22q^-90 + 18q^-89 + 32q^-88 + 54q^-87 + 17q^-86 + 24q^-85 - 86q^-84 - 160q^-83 - 102q^-82 - 15q^-81 + 166q^-80 + 225q^-79 + 389q^-78 + 110q^-77 - 256q^-76 - 520q^-75 - 642q^-74 - 356q^-73 + 22q^-72 + 997q^-71 + 1172q^-70 + 900q^-69 + 74q^-68 - 1032q^-67 - 1818q^-66 - 2135q^-65 - 477q^-64 + 1015q^-63 + 2648q^-62 + 3043q^-61 + 2026q^-60 - 444q^-59 - 3701q^-58 - 4309q^-57 - 3828q^-56 - 539q^-55 + 3254q^-54 + 6486q^-53 + 6356q^-52 + 2047q^-51 - 2526q^-50 - 7892q^-49 - 8909q^-48 - 5920q^-47 + 1870q^-46 + 9093q^-45 + 11599q^-44 + 9795q^-43 + 653q^-42 - 9093q^-41 - 16202q^-40 - 13447q^-39 - 3753q^-38 + 8574q^-37 + 19433q^-36 + 18881q^-35 + 8435q^-34 - 9668q^-33 - 21983q^-32 - 24182q^-31 - 13154q^-30 + 8683q^-29 + 26259q^-28 + 30926q^-27 + 15632q^-26 - 7793q^-25 - 30257q^-24 - 37102q^-23 - 20087q^-22 + 9674q^-21 + 36660q^-20 + 40860q^-19 + 23013q^-18 - 12262q^-17 - 43104q^-16 - 47116q^-15 - 21990q^-14 + 19289q^-13 + 48281q^-12 + 51100q^-11 + 19166q^-10 - 27811q^-9 - 57425q^-8 - 50697q^-7 - 9827q^-6 + 36874q^-5 + 64244q^-4 + 47771q^-3 - 2564q^-2 - 51600q^-1 - 66601 - 36353q + 17438q² + 64028q³ + 65590q⁴ + 20066q⁵ - 39619q⁶ - 71787q⁷ - 53786q⁸ + 948q⁹ + 59566q¹⁰ + 75008q¹¹ + 34853q¹² - 30977q¹³ - 74653q¹⁴ - 64998q¹⁵ - 8820q¹⁶ + 58906q¹⁷ + 84006q¹⁸ + 45793q¹⁹ - 27420q²⁰ - 81755q²¹ - 78153q²² - 17903q²³ + 61125q²⁴ + 97531q²⁵ + 61840q²⁶ - 20008q²⁷ - 88864q²⁸ - 96530q²⁹ - 36325q³⁰ + 54096q³¹ + 107686q³² + 84205q³³ + 1832q³⁴ - 80956q³⁵ - 108472q³⁶ - 62409q³⁷ + 27996q³⁸ + 97537q³⁹ + 97465q⁴⁰ + 32363q⁴¹ - 50565q⁴² - 96492q⁴³ - 77209q⁴⁴ - 6489q⁴⁵ + 63530q⁴⁶ + 84951q⁴⁷ + 49816q⁴⁸ - 13345q⁴⁹ - 61777q⁵⁰ - 66239q⁵¹ - 26293q⁵² + 25440q⁵³ + 52636q⁵⁴ + 42949q⁵⁵ + 8130q⁵⁶ - 26368q⁵⁷ - 39357q⁵⁸ - 24102q⁵⁹ + 3544q⁶⁰ + 22592q⁶¹ + 23970q⁶² + 10371q⁶³ - 6740q⁶⁴ - 16513q⁶⁵ - 12759q⁶⁶ - 1874q⁶⁷ + 6757q⁶⁸ + 9172q⁶⁹ + 5346q⁷⁰ - 720q⁷¹ - 5207q⁷² - 4487q⁷³ - 1118q⁷⁴ + 1501q⁷⁵ + 2547q⁷⁶ + 1687q⁷⁷ + 122q⁷⁸ - 1401q⁷⁹ - 1128q⁸⁰ - 209q⁸¹ + 269q⁸² + 542q⁸³ + 356q⁸⁴ + 86q⁸⁵ - 366q⁸⁶ - 203q⁸⁷ + 30q⁸⁸ + 27q⁸⁹ + 87q⁹⁰ + 39q⁹¹ + 41q⁹² - 95q⁹³ - 14q⁹⁴ + 33q⁹⁵ - 12q⁹⁶ + 10q⁹⁷ - 9q⁹⁸ + 19q⁹⁹ - 22q¹⁰⁰ + 5q¹⁰¹ + 12q¹⁰² - 9q¹⁰³ + 3q¹⁰⁴ - 6q¹⁰⁵ + 5q¹⁰⁶ - 2q¹⁰⁷ + q¹⁰⁸ + 2q¹⁰⁹ - 3q¹¹⁰ + q¹¹¹

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[10, 108]]
Out[2]=	PD[X[6, 2, 7, 1], X[16, 4, 17, 3], X[20, 13, 1, 14], X[14, 7, 15, 8], > X[8, 19, 9, 20], X[18, 9, 19, 10], X[10, 17, 11, 18], X[12, 6, 13, 5], > X[4, 12, 5, 11], X[2, 16, 3, 15]]
In[3]:=	GaussCode[Knot[10, 108]]
Out[3]=	GaussCode[1, -10, 2, -9, 8, -1, 4, -5, 6, -7, 9, -8, 3, -4, 10, -2, 7, -6, 5, > -3]
In[4]:=	DTCode[Knot[10, 108]]
Out[4]=	DTCode[6, 16, 12, 14, 18, 4, 20, 2, 10, 8]
In[5]:=	br = BR[Knot[10, 108]]
Out[5]=	BR[4, {1, 1, -2, 1, 1, 3, -2, 1, -2, -3, -3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 11}
In[7]:=	BraidIndex[Knot[10, 108]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[10, 108]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 108]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 3, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 108]][t]
Out[10]=	2 8 14 2 3 -15 + -- - -- + -- + 14 t - 8 t + 2 t 3 2 t t t
In[11]:=	Conway[Knot[10, 108]][z]
Out[11]=	4 6 1 + 4 z + 2 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 108], Knot[11, NonAlternating, 161]}
In[13]:=	{KnotDet[Knot[10, 108]], KnotSignature[Knot[10, 108]]}
Out[13]=	{63, 2}
In[14]:=	Jones[Knot[10, 108]][q]
Out[14]=	-4 3 5 8 2 3 4 5 6 -9 - q + -- - -- + - + 10 q - 10 q + 8 q - 5 q + 3 q - q 3 2 q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 108]}
In[16]:=	A2Invariant[Knot[10, 108]][q]
Out[16]=	-12 -10 2 -2 4 6 8 10 16 18 2 - q + q + -- - q - q + q - 2 q + 2 q + q - q 4 q
In[17]:=	HOMFLYPT[Knot[10, 108]][a, z]
Out[17]=	2 2 4 4 6 2 2 z 2 z 2 2 4 z 3 z 2 4 6 z 1 + 2 z - ---- + ---- - 2 a z + 3 z - -- + ---- - a z + z + -- 4 2 4 2 2 a a a a a
In[18]:=	Kauffman[Knot[10, 108]][a, z]
Out[18]=	2 2 3 3 2 z 6 z 3 2 z 2 z 2 2 z 3 z 1 - --- - --- - 6 a z - 2 a z - 10 z - -- + ---- - 7 a z + -- - ---- + 3 a 6 4 7 5 a a a a a 3 3 4 4 4 10 z 28 z 3 3 3 4 3 z 9 z 4 z 2 4 > ----- + ----- + 19 a z + 5 a z + 33 z + ---- - ---- + ---- + 17 a z + 3 a 6 4 2 a a a a 5 5 5 6 6 5 z 17 z 29 z 5 3 5 6 7 z 13 z > ---- - ----- - ----- - 11 a z - 4 a z - 33 z + ---- - ----- - 5 3 a 4 2 a a a a 7 7 8 9 2 6 8 z 4 z 7 3 7 8 6 z 2 8 2 z > 13 a z + ---- + ---- - 3 a z + a z + 9 z + ---- + 3 a z + ---- + 3 a 2 a a a 9 > 2 a z
In[19]:=	{Vassiliev[2][Knot[10, 108]], Vassiliev[3][Knot[10, 108]]}
Out[19]=	{0, 0}
In[20]:=	Kh[Knot[10, 108]][q, t]
Out[20]=	3 1 2 1 3 2 5 3 4 5 q 6 q + 5 q + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + --- + 9 5 7 4 5 4 5 3 3 3 3 2 2 q t t q t q t q t q t q t q t q t 3 5 5 2 7 2 7 3 9 3 9 4 > 5 q t + 5 q t + 3 q t + 5 q t + 2 q t + 3 q t + q t + 11 4 13 5 > 2 q t + q t
In[21]:=	ColouredJones[Knot[10, 108], 2][q]
Out[21]=	-13 3 -11 11 9 14 30 5 39 43 11 62 42 32 + q - --- - q + --- - -- - -- + -- - -- - -- + -- + -- - -- + -- - 12 10 9 8 7 6 5 4 3 2 q q q q q q q q q q q 2 3 4 5 6 7 8 9 > 73 q + 30 q + 47 q - 67 q + 13 q + 47 q - 47 q + 2 q + 30 q - 10 11 12 13 14 15 16 17 > 25 q + 3 q + 11 q - 11 q + 4 q + 2 q - 3 q + q

The Alternating Knot 10108

The Alternating Knot 10₁₀₈