The Alternating Knot 10₃₀

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Acknowledgement

KnotPlot

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

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

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

Minimum Braid Representative:

Length is 12, width is 5
Braid index is 5

A Morse Link Presentation:

3D Invariants:

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

Reversible 1 2 2 / NotAvailable 1

Alexander Polynomial: - 4t^-2 + 17t^-1 - 25 + 17t - 4t²

Conway Polynomial: 1 + z² - 4z⁴

Other knots with the same Alexander/Conway Polynomial: {K11a154, ...}

Determinant and Signature: {67, -2}

Jones Polynomial: q^-9 - 3q^-8 + 5q^-7 - 8q^-6 + 10q^-5 - 11q^-4 + 11q^-3 - 8q^-2 + 6q^-1 - 3 + q

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

A2 (sl(3)) Invariant: q^-28 - q^-26 - q^-24 + 2q^-22 - 2q^-20 + q^-16 - 2q^-14 + q^-12 - q^-10 + 2q^-8 + 2q^-6 - q^-4 + 3q^-2 - 1 - q² + q⁴

HOMFLY-PT Polynomial: z² + 2a² + a²z² - a²z⁴ - a⁴ - 2a⁴z² - 2a⁴z⁴ - a⁶z⁴ + a⁸z²

Kauffman Polynomial: - z² + z⁴ - 3az³ + 3az⁵ - 2a² + 5a²z² - 7a²z⁴ + 5a²z⁶ - a³z + 4a³z³ - 6a³z⁵ + 5a³z⁷ - a⁴ + 9a⁴z² - 11a⁴z⁴ + 2a⁴z⁶ + 3a⁴z⁸ - 5a⁵z + 16a⁵z³ - 19a⁵z⁵ + 7a⁵z⁷ + a⁵z⁹ + 2a⁶z² + 2a⁶z⁴ - 11a⁶z⁶ + 6a⁶z⁸ - 6a⁷z + 18a⁷z³ - 20a⁷z⁵ + 5a⁷z⁷ + a⁷z⁹ + a⁸z² + 2a⁸z⁴ - 7a⁸z⁶ + 3a⁸z⁸ - 2a⁹z + 9a⁹z³ - 10a⁹z⁵ + 3a⁹z⁷ + 2a¹⁰z² - 3a¹⁰z⁴ + a¹⁰z⁶

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=-2 is the signature of 10₃₀. 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 = 3 1

j = 1 2

j = -1 4 1

j = -3 5 3

j = -5 6 3

j = -7 5 5

j = -9 5 6

j = -11 3 5

j = -13 2 5

j = -15 1 3

j = -17 2

j = -19 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-26 - 3q^-25 + 10q^-23 - 12q^-22 - 8q^-21 + 32q^-20 - 20q^-19 - 30q^-18 + 61q^-17 - 20q^-16 - 61q^-15 + 84q^-14 - 9q^-13 - 87q^-12 + 91q^-11 + 6q^-10 - 94q^-9 + 77q^-8 + 15q^-7 - 74q^-6 + 49q^-5 + 14q^-4 - 42q^-3 + 23q^-2 + 7q^-1 - 16 + 7q + 2q² - 3q³ + q⁴

3 q^-51 - 3q^-50 + 5q^-48 + 5q^-47 - 12q^-46 - 14q^-45 + 19q^-44 + 31q^-43 - 22q^-42 - 58q^-41 + 17q^-40 + 92q^-39 + 2q^-38 - 131q^-37 - 32q^-36 + 159q^-35 + 84q^-34 - 187q^-33 - 135q^-32 + 192q^-31 + 201q^-30 - 191q^-29 - 258q^-28 + 170q^-27 + 317q^-26 - 144q^-25 - 364q^-24 + 108q^-23 + 398q^-22 - 67q^-21 - 420q^-20 + 30q^-19 + 413q^-18 + 17q^-17 - 401q^-16 - 36q^-15 + 346q^-14 + 69q^-13 - 301q^-12 - 67q^-11 + 229q^-10 + 73q^-9 - 177q^-8 - 53q^-7 + 118q^-6 + 45q^-5 - 83q^-4 - 26q^-3 + 50q^-2 + 18q^-1 - 32 - 9q + 18q² + 5q³ - 10q⁴ - q⁵ + 3q⁶ + 2q⁷ - 3q⁸ + q⁹

4 q^-84 - 3q^-83 + 5q^-81 + 5q^-79 - 19q^-78 - 7q^-77 + 20q^-76 + 11q^-75 + 37q^-74 - 60q^-73 - 55q^-72 + 22q^-71 + 40q^-70 + 155q^-69 - 85q^-68 - 161q^-67 - 70q^-66 + 17q^-65 + 394q^-64 + 16q^-63 - 236q^-62 - 294q^-61 - 201q^-60 + 636q^-59 + 286q^-58 - 94q^-57 - 516q^-56 - 669q^-55 + 672q^-54 + 579q^-53 + 330q^-52 - 520q^-51 - 1228q^-50 + 414q^-49 + 686q^-48 + 902q^-47 - 238q^-46 - 1669q^-45 - 27q^-44 + 554q^-43 + 1431q^-42 + 214q^-41 - 1904q^-40 - 499q^-39 + 269q^-38 + 1824q^-37 + 691q^-36 - 1945q^-35 - 906q^-34 - 79q^-33 + 2023q^-32 + 1110q^-31 - 1774q^-30 - 1173q^-29 - 448q^-28 + 1946q^-27 + 1379q^-26 - 1371q^-25 - 1180q^-24 - 757q^-23 + 1542q^-22 + 1377q^-21 - 831q^-20 - 891q^-19 - 865q^-18 + 958q^-17 + 1072q^-16 - 382q^-15 - 451q^-14 - 711q^-13 + 452q^-12 + 631q^-11 - 155q^-10 - 110q^-9 - 429q^-8 + 175q^-7 + 282q^-6 - 90q^-5 + 28q^-4 - 198q^-3 + 71q^-2 + 104q^-1 - 63 + 37q - 75q² + 33q³ + 35q⁴ - 34q⁵ + 18q⁶ - 24q⁷ + 12q⁸ + 11q⁹ - 12q¹⁰ + 5q¹¹ - 5q¹² + 3q¹³ + 2q¹⁴ - 3q¹⁵ + q¹⁶

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, 30]]

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

In[3]:=
GaussCode[Knot[10, 30]]

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

In[4]:=
DTCode[Knot[10, 30]]

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

In[5]:=
br = BR[Knot[10, 30]]

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

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

Out[6]=
{5, 12}

In[7]:=
BraidIndex[Knot[10, 30]]

Out[7]=
5

In[8]:=
Show[DrawMorseLink[Knot[10, 30]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[10, 30]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

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

In[10]:=
alex = Alexander[Knot[10, 30]][t]

Out[10]=
4 17 2 -25 - -- + -- + 17 t - 4 t 2 t t

In[11]:=
Conway[Knot[10, 30]][z]

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

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

Out[12]=
{Knot[10, 30], Knot[11, Alternating, 154]}

In[13]:=
{KnotDet[Knot[10, 30]], KnotSignature[Knot[10, 30]]}

Out[13]=
{67, -2}

In[14]:=
Jones[Knot[10, 30]][q]

Out[14]=
-9 3 5 8 10 11 11 8 6 -3 + q - -- + -- - -- + -- - -- + -- - -- + - + q 8 7 6 5 4 3 2 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, 30]}

In[16]:=
A2Invariant[Knot[10, 30]][q]

Out[16]=
-28 -26 -24 2 2 -16 2 -12 -10 2 2 -1 + q - q - q + --- - --- + q - --- + q - q + -- + -- - 22 20 14 8 6 q q q q q -4 3 2 4 > q + -- - q + q 2 q

In[17]:=
HOMFLYPT[Knot[10, 30]][a, z]

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

In[18]:=
Kauffman[Knot[10, 30]][a, z]

Out[18]=
2 4 3 5 7 9 2 2 2 4 2 -2 a - a - a z - 5 a z - 6 a z - 2 a z - z + 5 a z + 9 a z + 6 2 8 2 10 2 3 3 3 5 3 7 3 > 2 a z + a z + 2 a z - 3 a z + 4 a z + 16 a z + 18 a z + 9 3 4 2 4 4 4 6 4 8 4 10 4 5 > 9 a z + z - 7 a z - 11 a z + 2 a z + 2 a z - 3 a z + 3 a z - 3 5 5 5 7 5 9 5 2 6 4 6 6 6 > 6 a z - 19 a z - 20 a z - 10 a z + 5 a z + 2 a z - 11 a z - 8 6 10 6 3 7 5 7 7 7 9 7 4 8 > 7 a z + a z + 5 a z + 7 a z + 5 a z + 3 a z + 3 a z + 6 8 8 8 5 9 7 9 > 6 a z + 3 a z + a z + a z

In[19]:=
{Vassiliev[2][Knot[10, 30]], Vassiliev[3][Knot[10, 30]]}

Out[19]=
{1, -1}

In[20]:=
Kh[Knot[10, 30]][q, t]

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

In[21]:=
ColouredJones[Knot[10, 30], 2][q]

Out[21]=
-26 3 10 12 8 32 20 30 61 20 61 84 -16 + q - --- + --- - --- - --- + --- - --- - --- + --- - --- - --- + --- - 25 23 22 21 20 19 18 17 16 15 14 q q q q q q q q q q q 9 87 91 6 94 77 15 74 49 14 42 23 7 > --- - --- + --- + --- - -- + -- + -- - -- + -- + -- - -- + -- + - + 7 q + 13 12 11 10 9 8 7 6 5 4 3 2 q q q q q q q q q q q q q 2 3 4 > 2 q - 3 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10₃₀

10₂₉
10₃₁

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-26 - 3q^-25 + 10q^-23 - 12q^-22 - 8q^-21 + 32q^-20 - 20q^-19 - 30q^-18 + 61q^-17 - 20q^-16 - 61q^-15 + 84q^-14 - 9q^-13 - 87q^-12 + 91q^-11 + 6q^-10 - 94q^-9 + 77q^-8 + 15q^-7 - 74q^-6 + 49q^-5 + 14q^-4 - 42q^-3 + 23q^-2 + 7q^-1 - 16 + 7q + 2q² - 3q³ + q⁴
3	q^-51 - 3q^-50 + 5q^-48 + 5q^-47 - 12q^-46 - 14q^-45 + 19q^-44 + 31q^-43 - 22q^-42 - 58q^-41 + 17q^-40 + 92q^-39 + 2q^-38 - 131q^-37 - 32q^-36 + 159q^-35 + 84q^-34 - 187q^-33 - 135q^-32 + 192q^-31 + 201q^-30 - 191q^-29 - 258q^-28 + 170q^-27 + 317q^-26 - 144q^-25 - 364q^-24 + 108q^-23 + 398q^-22 - 67q^-21 - 420q^-20 + 30q^-19 + 413q^-18 + 17q^-17 - 401q^-16 - 36q^-15 + 346q^-14 + 69q^-13 - 301q^-12 - 67q^-11 + 229q^-10 + 73q^-9 - 177q^-8 - 53q^-7 + 118q^-6 + 45q^-5 - 83q^-4 - 26q^-3 + 50q^-2 + 18q^-1 - 32 - 9q + 18q² + 5q³ - 10q⁴ - q⁵ + 3q⁶ + 2q⁷ - 3q⁸ + q⁹
4	q^-84 - 3q^-83 + 5q^-81 + 5q^-79 - 19q^-78 - 7q^-77 + 20q^-76 + 11q^-75 + 37q^-74 - 60q^-73 - 55q^-72 + 22q^-71 + 40q^-70 + 155q^-69 - 85q^-68 - 161q^-67 - 70q^-66 + 17q^-65 + 394q^-64 + 16q^-63 - 236q^-62 - 294q^-61 - 201q^-60 + 636q^-59 + 286q^-58 - 94q^-57 - 516q^-56 - 669q^-55 + 672q^-54 + 579q^-53 + 330q^-52 - 520q^-51 - 1228q^-50 + 414q^-49 + 686q^-48 + 902q^-47 - 238q^-46 - 1669q^-45 - 27q^-44 + 554q^-43 + 1431q^-42 + 214q^-41 - 1904q^-40 - 499q^-39 + 269q^-38 + 1824q^-37 + 691q^-36 - 1945q^-35 - 906q^-34 - 79q^-33 + 2023q^-32 + 1110q^-31 - 1774q^-30 - 1173q^-29 - 448q^-28 + 1946q^-27 + 1379q^-26 - 1371q^-25 - 1180q^-24 - 757q^-23 + 1542q^-22 + 1377q^-21 - 831q^-20 - 891q^-19 - 865q^-18 + 958q^-17 + 1072q^-16 - 382q^-15 - 451q^-14 - 711q^-13 + 452q^-12 + 631q^-11 - 155q^-10 - 110q^-9 - 429q^-8 + 175q^-7 + 282q^-6 - 90q^-5 + 28q^-4 - 198q^-3 + 71q^-2 + 104q^-1 - 63 + 37q - 75q² + 33q³ + 35q⁴ - 34q⁵ + 18q⁶ - 24q⁷ + 12q⁸ + 11q⁹ - 12q¹⁰ + 5q¹¹ - 5q¹² + 3q¹³ + 2q¹⁴ - 3q¹⁵ + q¹⁶

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

Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 30]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 1, 2, 2, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 30]][t]
Out[10]=	4 17 2 -25 - -- + -- + 17 t - 4 t 2 t t
In[11]:=	Conway[Knot[10, 30]][z]
Out[11]=	2 4 1 + z - 4 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 30], Knot[11, Alternating, 154]}
In[13]:=	{KnotDet[Knot[10, 30]], KnotSignature[Knot[10, 30]]}
Out[13]=	{67, -2}
In[14]:=	Jones[Knot[10, 30]][q]
Out[14]=	-9 3 5 8 10 11 11 8 6 -3 + q - -- + -- - -- + -- - -- + -- - -- + - + q 8 7 6 5 4 3 2 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, 30]}
In[16]:=	A2Invariant[Knot[10, 30]][q]
Out[16]=	-28 -26 -24 2 2 -16 2 -12 -10 2 2 -1 + q - q - q + --- - --- + q - --- + q - q + -- + -- - 22 20 14 8 6 q q q q q -4 3 2 4 > q + -- - q + q 2 q
In[17]:=	HOMFLYPT[Knot[10, 30]][a, z]
Out[17]=	2 4 2 2 2 4 2 8 2 2 4 4 4 6 4 2 a - a + z + a z - 2 a z + a z - a z - 2 a z - a z
In[18]:=	Kauffman[Knot[10, 30]][a, z]
Out[18]=	2 4 3 5 7 9 2 2 2 4 2 -2 a - a - a z - 5 a z - 6 a z - 2 a z - z + 5 a z + 9 a z + 6 2 8 2 10 2 3 3 3 5 3 7 3 > 2 a z + a z + 2 a z - 3 a z + 4 a z + 16 a z + 18 a z + 9 3 4 2 4 4 4 6 4 8 4 10 4 5 > 9 a z + z - 7 a z - 11 a z + 2 a z + 2 a z - 3 a z + 3 a z - 3 5 5 5 7 5 9 5 2 6 4 6 6 6 > 6 a z - 19 a z - 20 a z - 10 a z + 5 a z + 2 a z - 11 a z - 8 6 10 6 3 7 5 7 7 7 9 7 4 8 > 7 a z + a z + 5 a z + 7 a z + 5 a z + 3 a z + 3 a z + 6 8 8 8 5 9 7 9 > 6 a z + 3 a z + a z + a z
In[19]:=	{Vassiliev[2][Knot[10, 30]], Vassiliev[3][Knot[10, 30]]}
Out[19]=	{1, -1}
In[20]:=	Kh[Knot[10, 30]][q, t]
Out[20]=	3 4 1 2 1 3 2 5 3 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 3 q 19 8 17 7 15 7 15 6 13 6 13 5 11 5 q q t q t q t q t q t q t q t 5 5 6 5 5 6 3 5 t > ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- + - + 2 q t + 11 4 9 4 9 3 7 3 7 2 5 2 5 3 q q t q t q t q t q t q t q t q t 3 2 > q t
In[21]:=	ColouredJones[Knot[10, 30], 2][q]
Out[21]=	-26 3 10 12 8 32 20 30 61 20 61 84 -16 + q - --- + --- - --- - --- + --- - --- - --- + --- - --- - --- + --- - 25 23 22 21 20 19 18 17 16 15 14 q q q q q q q q q q q 9 87 91 6 94 77 15 74 49 14 42 23 7 > --- - --- + --- + --- - -- + -- + -- - -- + -- + -- - -- + -- + - + 7 q + 13 12 11 10 9 8 7 6 5 4 3 2 q q q q q q q q q q q q q 2 3 4 > 2 q - 3 q + q

The Alternating Knot 1030

The Alternating Knot 10₃₀