The Alternating Knot 10₃₉

Visit 10₃₉'s page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 10₃₉'s page at Knotilus!

Acknowledgement

KnotPlot

PD Presentation: X₁₄₂₅ X_5,12,6,13 X_3,11,4,10 X_11,3,12,2 X_7,14,8,15 X_9,18,10,19 X_15,20,16,1 X_19,16,20,17 X_13,6,14,7 X_17,8,18,9

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

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

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 2 / NotAvailable 1

Alexander Polynomial: - 2t^-3 + 8t^-2 - 13t^-1 + 15 - 13t + 8t² - 2t³

Conway Polynomial: 1 + z² - 4z⁴ - 2z⁶

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

Determinant and Signature: {61, -4}

Jones Polynomial: q^-10 - 3q^-9 + 5q^-8 - 8q^-7 + 10q^-6 - 10q^-5 + 9q^-4 - 7q^-3 + 5q^-2 - 2q^-1 + 1

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

A2 (sl(3)) Invariant: q^-30 - q^-28 - 2q^-22 + 2q^-20 - q^-18 + q^-16 - 2q^-12 + 2q^-10 - q^-8 + 2q^-6 + q^-4 + 1

HOMFLY-PT Polynomial: 2a² + 3a²z² + a²z⁴ - a⁴ - 2a⁴z² - 3a⁴z⁴ - a⁴z⁶ - 2a⁶z² - 3a⁶z⁴ - a⁶z⁶ + 2a⁸z² + a⁸z⁴

Kauffman Polynomial: - 2a² + 5a²z² - 4a²z⁴ + a²z⁶ + 4a³z³ - 6a³z⁵ + 2a³z⁷ - a⁴ + 5a⁴z² - 4a⁴z⁴ - 3a⁴z⁶ + 2a⁴z⁸ - a⁵z + 5a⁵z³ - 9a⁵z⁵ + 2a⁵z⁷ + a⁵z⁹ - a⁶z² + 5a⁶z⁴ - 10a⁶z⁶ + 5a⁶z⁸ + 4a⁷z³ - 9a⁷z⁵ + 4a⁷z⁷ + a⁷z⁹ + a⁸z² - 2a⁸z⁶ + 3a⁸z⁸ + 2a⁹z - a⁹z³ - 3a⁹z⁵ + 4a⁹z⁷ + a¹⁰z² - 4a¹⁰z⁴ + 4a¹⁰z⁶ + a¹¹z - 4a¹¹z³ + 3a¹¹z⁵ - a¹²z² + a¹²z⁴

V₂ and V₃, the type 2 and 3 Vassiliev invariants: {1, -1}

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=-4 is the signature of 10₃₉. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

t^rq^j r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2

j = 1 1

j = -1 1

j = -3 4 1

j = -5 4 2

j = -7 5 3

j = -9 5 4

j = -11 5 5

j = -13 3 5

j = -15 2 5

j = -17 1 3

j = -19 2

j = -21 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-28 - 3q^-27 + q^-26 + 7q^-25 - 13q^-24 + 5q^-23 + 20q^-22 - 36q^-21 + 10q^-20 + 44q^-19 - 64q^-18 + 10q^-17 + 69q^-16 - 79q^-15 + q^-14 + 79q^-13 - 72q^-12 - 11q^-11 + 72q^-10 - 49q^-9 - 19q^-8 + 51q^-7 - 22q^-6 - 18q^-5 + 25q^-4 - 5q^-3 - 9q^-2 + 7q^-1 - 2q + q²

3 q^-54 - 3q^-53 + q^-52 + 3q^-51 + 2q^-50 - 8q^-49 - q^-48 + 13q^-47 - 3q^-46 - 20q^-45 + 7q^-44 + 38q^-43 - 20q^-42 - 60q^-41 + 30q^-40 + 100q^-39 - 45q^-38 - 144q^-37 + 46q^-36 + 203q^-35 - 47q^-34 - 253q^-33 + 32q^-32 + 297q^-31 - 7q^-30 - 329q^-29 - 19q^-28 + 339q^-27 + 52q^-26 - 336q^-25 - 82q^-24 + 314q^-23 + 114q^-22 - 284q^-21 - 138q^-20 + 239q^-19 + 160q^-18 - 193q^-17 - 164q^-16 + 133q^-15 + 168q^-14 - 88q^-13 - 146q^-12 + 37q^-11 + 126q^-10 - 10q^-9 - 87q^-8 - 17q^-7 + 65q^-6 + 16q^-5 - 32q^-4 - 20q^-3 + 19q^-2 + 11q^-1 - 6 - 8q + 4q² + 2q³ - 2q⁵ + q⁶

4 q^-88 - 3q^-87 + q^-86 + 3q^-85 - 2q^-84 + 7q^-83 - 14q^-82 + 3q^-81 + 7q^-80 - 12q^-79 + 29q^-78 - 31q^-77 + 15q^-76 + 11q^-75 - 55q^-74 + 55q^-73 - 42q^-72 + 76q^-71 + 33q^-70 - 171q^-69 + 30q^-68 - 57q^-67 + 253q^-66 + 148q^-65 - 369q^-64 - 139q^-63 - 153q^-62 + 565q^-61 + 454q^-60 - 551q^-59 - 468q^-58 - 443q^-57 + 890q^-56 + 946q^-55 - 571q^-54 - 812q^-53 - 901q^-52 + 1043q^-51 + 1424q^-50 - 390q^-49 - 968q^-48 - 1351q^-47 + 956q^-46 + 1687q^-45 - 115q^-44 - 889q^-43 - 1623q^-42 + 709q^-41 + 1677q^-40 + 152q^-39 - 637q^-38 - 1690q^-37 + 377q^-36 + 1456q^-35 + 383q^-34 - 284q^-33 - 1581q^-32 + 10q^-31 + 1074q^-30 + 544q^-29 + 113q^-28 - 1298q^-27 - 302q^-26 + 589q^-25 + 548q^-24 + 444q^-23 - 854q^-22 - 433q^-21 + 125q^-20 + 364q^-19 + 571q^-18 - 384q^-17 - 338q^-16 - 145q^-15 + 104q^-14 + 453q^-13 - 68q^-12 - 136q^-11 - 176q^-10 - 59q^-9 + 234q^-8 + 31q^-7 - q^-6 - 88q^-5 - 76q^-4 + 77q^-3 + 17q^-2 + 26q^-1 - 21 - 36q + 19q² + 11q⁴ - 2q⁵ - 10q⁶ + 5q⁷ - q⁸ + 2q⁹ - 2q¹¹ + q¹²

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, 39]]

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

In[3]:=
GaussCode[Knot[10, 39]]

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

In[4]:=
DTCode[Knot[10, 39]]

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

In[5]:=
br = BR[Knot[10, 39]]

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

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

Out[6]=
{4, 11}

In[7]:=
BraidIndex[Knot[10, 39]]

Out[7]=
4

In[8]:=
Show[DrawMorseLink[Knot[10, 39]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[10, 39]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

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

In[10]:=
alex = Alexander[Knot[10, 39]][t]

Out[10]=
2 8 13 2 3 15 - -- + -- - -- - 13 t + 8 t - 2 t 3 2 t t t

In[11]:=
Conway[Knot[10, 39]][z]

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

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

Out[12]=
{Knot[10, 39]}

In[13]:=
{KnotDet[Knot[10, 39]], KnotSignature[Knot[10, 39]]}

Out[13]=
{61, -4}

In[14]:=
Jones[Knot[10, 39]][q]

Out[14]=
-10 3 5 8 10 10 9 7 5 2 1 + q - -- + -- - -- + -- - -- + -- - -- + -- - - 9 8 7 6 5 4 3 2 q q q q q q q q q

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

Out[15]=
{Knot[10, 39]}

In[16]:=
A2Invariant[Knot[10, 39]][q]

Out[16]=
-30 -28 2 2 -18 -16 2 2 -8 2 -4 1 + q - q - --- + --- - q + q - --- + --- - q + -- + q 22 20 12 10 6 q q q q q

In[17]:=
HOMFLYPT[Knot[10, 39]][a, z]

Out[17]=
2 4 2 2 4 2 6 2 8 2 2 4 4 4 6 4 2 a - a + 3 a z - 2 a z - 2 a z + 2 a z + a z - 3 a z - 3 a z + 8 4 4 6 6 6 > a z - a z - a z

In[18]:=
Kauffman[Knot[10, 39]][a, z]

Out[18]=
2 4 5 9 11 2 2 4 2 6 2 8 2 -2 a - a - a z + 2 a z + a z + 5 a z + 5 a z - a z + a z + 10 2 12 2 3 3 5 3 7 3 9 3 11 3 > a z - a z + 4 a z + 5 a z + 4 a z - a z - 4 a z - 2 4 4 4 6 4 10 4 12 4 3 5 5 5 > 4 a z - 4 a z + 5 a z - 4 a z + a z - 6 a z - 9 a z - 7 5 9 5 11 5 2 6 4 6 6 6 8 6 > 9 a z - 3 a z + 3 a z + a z - 3 a z - 10 a z - 2 a z + 10 6 3 7 5 7 7 7 9 7 4 8 6 8 > 4 a z + 2 a z + 2 a z + 4 a z + 4 a z + 2 a z + 5 a z + 8 8 5 9 7 9 > 3 a z + a z + a z

In[19]:=
{Vassiliev[2][Knot[10, 39]], Vassiliev[3][Knot[10, 39]]}

Out[19]=
{1, -1}

In[20]:=
Kh[Knot[10, 39]][q, t]

Out[20]=
2 4 1 2 1 3 2 5 3 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 5 3 21 8 19 7 17 7 17 6 15 6 15 5 13 5 q q q t q t q t q t q t q t q t 5 5 5 5 4 5 3 4 t t > ------ + ------ + ------ + ----- + ----- + ----- + ---- + ---- + -- + - + 13 4 11 4 11 3 9 3 9 2 7 2 7 5 3 q q t q t q t q t q t q t q t q t q 2 > q t

In[21]:=
ColouredJones[Knot[10, 39], 2][q]

Out[21]=
-28 3 -26 7 13 5 20 36 10 44 64 10 69 q - --- + q + --- - --- + --- + --- - --- + --- + --- - --- + --- + --- - 27 25 24 23 22 21 20 19 18 17 16 q q q q q q q q q q q 79 -14 79 72 11 72 49 19 51 22 18 25 5 > --- + q + --- - --- - --- + --- - -- - -- + -- - -- - -- + -- - -- - 15 13 12 11 10 9 8 7 6 5 4 3 q q q q q q q q q q q q 9 7 2 > -- + - - 2 q + q 2 q q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10₃₉

10₃₈
10₄₀

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-28 - 3q^-27 + q^-26 + 7q^-25 - 13q^-24 + 5q^-23 + 20q^-22 - 36q^-21 + 10q^-20 + 44q^-19 - 64q^-18 + 10q^-17 + 69q^-16 - 79q^-15 + q^-14 + 79q^-13 - 72q^-12 - 11q^-11 + 72q^-10 - 49q^-9 - 19q^-8 + 51q^-7 - 22q^-6 - 18q^-5 + 25q^-4 - 5q^-3 - 9q^-2 + 7q^-1 - 2q + q²
3	q^-54 - 3q^-53 + q^-52 + 3q^-51 + 2q^-50 - 8q^-49 - q^-48 + 13q^-47 - 3q^-46 - 20q^-45 + 7q^-44 + 38q^-43 - 20q^-42 - 60q^-41 + 30q^-40 + 100q^-39 - 45q^-38 - 144q^-37 + 46q^-36 + 203q^-35 - 47q^-34 - 253q^-33 + 32q^-32 + 297q^-31 - 7q^-30 - 329q^-29 - 19q^-28 + 339q^-27 + 52q^-26 - 336q^-25 - 82q^-24 + 314q^-23 + 114q^-22 - 284q^-21 - 138q^-20 + 239q^-19 + 160q^-18 - 193q^-17 - 164q^-16 + 133q^-15 + 168q^-14 - 88q^-13 - 146q^-12 + 37q^-11 + 126q^-10 - 10q^-9 - 87q^-8 - 17q^-7 + 65q^-6 + 16q^-5 - 32q^-4 - 20q^-3 + 19q^-2 + 11q^-1 - 6 - 8q + 4q² + 2q³ - 2q⁵ + q⁶
4	q^-88 - 3q^-87 + q^-86 + 3q^-85 - 2q^-84 + 7q^-83 - 14q^-82 + 3q^-81 + 7q^-80 - 12q^-79 + 29q^-78 - 31q^-77 + 15q^-76 + 11q^-75 - 55q^-74 + 55q^-73 - 42q^-72 + 76q^-71 + 33q^-70 - 171q^-69 + 30q^-68 - 57q^-67 + 253q^-66 + 148q^-65 - 369q^-64 - 139q^-63 - 153q^-62 + 565q^-61 + 454q^-60 - 551q^-59 - 468q^-58 - 443q^-57 + 890q^-56 + 946q^-55 - 571q^-54 - 812q^-53 - 901q^-52 + 1043q^-51 + 1424q^-50 - 390q^-49 - 968q^-48 - 1351q^-47 + 956q^-46 + 1687q^-45 - 115q^-44 - 889q^-43 - 1623q^-42 + 709q^-41 + 1677q^-40 + 152q^-39 - 637q^-38 - 1690q^-37 + 377q^-36 + 1456q^-35 + 383q^-34 - 284q^-33 - 1581q^-32 + 10q^-31 + 1074q^-30 + 544q^-29 + 113q^-28 - 1298q^-27 - 302q^-26 + 589q^-25 + 548q^-24 + 444q^-23 - 854q^-22 - 433q^-21 + 125q^-20 + 364q^-19 + 571q^-18 - 384q^-17 - 338q^-16 - 145q^-15 + 104q^-14 + 453q^-13 - 68q^-12 - 136q^-11 - 176q^-10 - 59q^-9 + 234q^-8 + 31q^-7 - q^-6 - 88q^-5 - 76q^-4 + 77q^-3 + 17q^-2 + 26q^-1 - 21 - 36q + 19q² + 11q⁴ - 2q⁵ - 10q⁶ + 5q⁷ - q⁸ + 2q⁹ - 2q¹¹ + q¹²

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[10, 39]]
Out[2]=	PD[X[1, 4, 2, 5], X[5, 12, 6, 13], X[3, 11, 4, 10], X[11, 3, 12, 2], > X[7, 14, 8, 15], X[9, 18, 10, 19], X[15, 20, 16, 1], X[19, 16, 20, 17], > X[13, 6, 14, 7], X[17, 8, 18, 9]]
In[3]:=	GaussCode[Knot[10, 39]]
Out[3]=	GaussCode[-1, 4, -3, 1, -2, 9, -5, 10, -6, 3, -4, 2, -9, 5, -7, 8, -10, 6, -8, > 7]
In[4]:=	DTCode[Knot[10, 39]]
Out[4]=	DTCode[4, 10, 12, 14, 18, 2, 6, 20, 8, 16]
In[5]:=	br = BR[Knot[10, 39]]
Out[5]=	BR[4, {-1, -1, -1, -2, 1, -2, -2, -2, 3, -2, 3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 11}
In[7]:=	BraidIndex[Knot[10, 39]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[10, 39]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 39]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 3, 2, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 39]][t]
Out[10]=	2 8 13 2 3 15 - -- + -- - -- - 13 t + 8 t - 2 t 3 2 t t t
In[11]:=	Conway[Knot[10, 39]][z]
Out[11]=	2 4 6 1 + z - 4 z - 2 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 39]}
In[13]:=	{KnotDet[Knot[10, 39]], KnotSignature[Knot[10, 39]]}
Out[13]=	{61, -4}
In[14]:=	Jones[Knot[10, 39]][q]
Out[14]=	-10 3 5 8 10 10 9 7 5 2 1 + q - -- + -- - -- + -- - -- + -- - -- + -- - - 9 8 7 6 5 4 3 2 q q q q q q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 39]}
In[16]:=	A2Invariant[Knot[10, 39]][q]
Out[16]=	-30 -28 2 2 -18 -16 2 2 -8 2 -4 1 + q - q - --- + --- - q + q - --- + --- - q + -- + q 22 20 12 10 6 q q q q q
In[17]:=	HOMFLYPT[Knot[10, 39]][a, z]
Out[17]=	2 4 2 2 4 2 6 2 8 2 2 4 4 4 6 4 2 a - a + 3 a z - 2 a z - 2 a z + 2 a z + a z - 3 a z - 3 a z + 8 4 4 6 6 6 > a z - a z - a z
In[18]:=	Kauffman[Knot[10, 39]][a, z]
Out[18]=	2 4 5 9 11 2 2 4 2 6 2 8 2 -2 a - a - a z + 2 a z + a z + 5 a z + 5 a z - a z + a z + 10 2 12 2 3 3 5 3 7 3 9 3 11 3 > a z - a z + 4 a z + 5 a z + 4 a z - a z - 4 a z - 2 4 4 4 6 4 10 4 12 4 3 5 5 5 > 4 a z - 4 a z + 5 a z - 4 a z + a z - 6 a z - 9 a z - 7 5 9 5 11 5 2 6 4 6 6 6 8 6 > 9 a z - 3 a z + 3 a z + a z - 3 a z - 10 a z - 2 a z + 10 6 3 7 5 7 7 7 9 7 4 8 6 8 > 4 a z + 2 a z + 2 a z + 4 a z + 4 a z + 2 a z + 5 a z + 8 8 5 9 7 9 > 3 a z + a z + a z
In[19]:=	{Vassiliev[2][Knot[10, 39]], Vassiliev[3][Knot[10, 39]]}
Out[19]=	{1, -1}
In[20]:=	Kh[Knot[10, 39]][q, t]
Out[20]=	2 4 1 2 1 3 2 5 3 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 5 3 21 8 19 7 17 7 17 6 15 6 15 5 13 5 q q q t q t q t q t q t q t q t 5 5 5 5 4 5 3 4 t t > ------ + ------ + ------ + ----- + ----- + ----- + ---- + ---- + -- + - + 13 4 11 4 11 3 9 3 9 2 7 2 7 5 3 q q t q t q t q t q t q t q t q t q 2 > q t
In[21]:=	ColouredJones[Knot[10, 39], 2][q]
Out[21]=	-28 3 -26 7 13 5 20 36 10 44 64 10 69 q - --- + q + --- - --- + --- + --- - --- + --- + --- - --- + --- + --- - 27 25 24 23 22 21 20 19 18 17 16 q q q q q q q q q q q 79 -14 79 72 11 72 49 19 51 22 18 25 5 > --- + q + --- - --- - --- + --- - -- - -- + -- - -- - -- + -- - -- - 15 13 12 11 10 9 8 7 6 5 4 3 q q q q q q q q q q q q 9 7 2 > -- + - - 2 q + q 2 q q

The Alternating Knot 1039

The Alternating Knot 10₃₉