The Non Alternating Knot 10₁₃₁

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Acknowledgement

KnotPlot

PD Presentation: X₁₄₂₅ X₃₈₄₉ X_14,6,15,5 X_15,20,16,1 X_9,16,10,17 X_19,10,20,11 X_11,18,12,19 X_17,12,18,13 X_6,14,7,13 X₇₂₈₃

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

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

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 1 2 3 / NotAvailable 1

Alexander Polynomial: - 2t^-2 + 8t^-1 - 11 + 8t - 2t²

Conway Polynomial: 1 - 2z⁴

Other knots with the same Alexander/Conway Polynomial: {8₁₄, 9₈, ...}

Determinant and Signature: {31, -2}

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

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

A2 (sl(3)) Invariant: q^-28 + q^-22 - 2q^-20 - q^-18 - q^-16 - q^-14 + q^-12 + 2q^-8 + q^-6 + 2q^-2

HOMFLY-PT Polynomial: 2a² + 2a²z² - a⁴z² - a⁴z⁴ - 2a⁶ - 2a⁶z² - a⁶z⁴ + a⁸ + a⁸z²

Kauffman Polynomial: - 2a² + 3a²z² + a³z + a³z³ + a³z⁵ + 2a⁴z² - 2a⁴z⁴ + 2a⁴z⁶ - a⁵z + 2a⁵z³ - 3a⁵z⁵ + 2a⁵z⁷ + 2a⁶ - 3a⁶z² - 2a⁶z⁴ + a⁶z⁸ - 5a⁷z + 10a⁷z³ - 12a⁷z⁵ + 4a⁷z⁷ + a⁸ + 2a⁸z² - 4a⁸z⁴ - a⁸z⁶ + a⁸z⁸ - 3a⁹z + 9a⁹z³ - 8a⁹z⁵ + 2a⁹z⁷ + 4a¹⁰z² - 4a¹⁰z⁴ + a¹⁰z⁶

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

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 10₁₃₁. 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

j = -1 2

j = -3 2 1

j = -5 3 1

j = -7 2 2

j = -9 3 3

j = -11 2 2

j = -13 1 3

j = -15 1 2

j = -17 1

j = -19 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-26 - 2q^-25 - q^-24 + 6q^-23 - 4q^-22 - 7q^-21 + 13q^-20 - 2q^-19 - 16q^-18 + 18q^-17 + 3q^-16 - 23q^-15 + 17q^-14 + 10q^-13 - 26q^-12 + 12q^-11 + 14q^-10 - 23q^-9 + 7q^-8 + 11q^-7 - 13q^-6 + 3q^-5 + 5q^-4 - 4q^-3 + q^-2 + q^-1

3 q^-51 - 2q^-50 - q^-49 + 2q^-48 + 5q^-47 - 3q^-46 - 9q^-45 + 15q^-43 + 5q^-42 - 18q^-41 - 15q^-40 + 19q^-39 + 25q^-38 - 15q^-37 - 33q^-36 + 6q^-35 + 41q^-34 + q^-33 - 40q^-32 - 16q^-31 + 43q^-30 + 24q^-29 - 37q^-28 - 38q^-27 + 35q^-26 + 46q^-25 - 27q^-24 - 57q^-23 + 22q^-22 + 63q^-21 - 15q^-20 - 63q^-19 + 4q^-18 + 63q^-17 - q^-16 - 50q^-15 - 9q^-14 + 43q^-13 + 6q^-12 - 25q^-11 - 10q^-10 + 18q^-9 + 3q^-8 - 6q^-7 - 5q^-6 + 7q^-5 - 2q^-4 + q^-3 - 2q^-2 + 2q^-1

4 q^-84 - 2q^-83 - q^-82 + 2q^-81 + q^-80 + 6q^-79 - 7q^-78 - 7q^-77 + 25q^-74 - 3q^-73 - 15q^-72 - 16q^-71 - 21q^-70 + 46q^-69 + 21q^-68 + 5q^-67 - 27q^-66 - 73q^-65 + 32q^-64 + 36q^-63 + 58q^-62 + 8q^-61 - 112q^-60 - 17q^-59 - 5q^-58 + 95q^-57 + 85q^-56 - 94q^-55 - 50q^-54 - 89q^-53 + 74q^-52 + 151q^-51 - 29q^-50 - 38q^-49 - 168q^-48 + 12q^-47 + 180q^-46 + 44q^-45 + q^-44 - 222q^-43 - 57q^-42 + 186q^-41 + 112q^-40 + 39q^-39 - 262q^-38 - 117q^-37 + 181q^-36 + 172q^-35 + 76q^-34 - 282q^-33 - 174q^-32 + 154q^-31 + 211q^-30 + 119q^-29 - 256q^-28 - 207q^-27 + 92q^-26 + 192q^-25 + 150q^-24 - 170q^-23 - 184q^-22 + 21q^-21 + 115q^-20 + 132q^-19 - 72q^-18 - 108q^-17 - 11q^-16 + 34q^-15 + 73q^-14 - 17q^-13 - 37q^-12 - 5q^-11 + 22q^-9 - 4q^-8 - 7q^-7 + 2q^-6 - 2q^-5 + 4q^-4 - q^-3 - 2q^-2 + q^-1 + 1

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, 131]]

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

In[3]:=
GaussCode[Knot[10, 131]]

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

In[4]:=
DTCode[Knot[10, 131]]

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

In[5]:=
br = BR[Knot[10, 131]]

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

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

Out[6]=
{4, 11}

In[7]:=
BraidIndex[Knot[10, 131]]

Out[7]=
4

In[8]:=
Show[DrawMorseLink[Knot[10, 131]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[10, 131]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

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

In[10]:=
alex = Alexander[Knot[10, 131]][t]

Out[10]=
2 8 2 -11 - -- + - + 8 t - 2 t 2 t t

In[11]:=
Conway[Knot[10, 131]][z]

Out[11]=
4 1 - 2 z

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

Out[12]=
{Knot[8, 14], Knot[9, 8], Knot[10, 131]}

In[13]:=
{KnotDet[Knot[10, 131]], KnotSignature[Knot[10, 131]]}

Out[13]=
{31, -2}

In[14]:=
Jones[Knot[10, 131]][q]

Out[14]=
-9 2 3 5 5 5 5 3 2 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, 131]}

In[16]:=
A2Invariant[Knot[10, 131]][q]

Out[16]=
-28 -22 2 -18 -16 -14 -12 2 -6 2 q + q - --- - q - q - q + q + -- + q + -- 20 8 2 q q q

In[17]:=
HOMFLYPT[Knot[10, 131]][a, z]

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

In[18]:=
Kauffman[Knot[10, 131]][a, z]

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

In[19]:=
{Vassiliev[2][Knot[10, 131]], Vassiliev[3][Knot[10, 131]]}

Out[19]=
{0, 2}

In[20]:=
Kh[Knot[10, 131]][q, t]

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

In[21]:=
ColouredJones[Knot[10, 131], 2][q]

Out[21]=
-26 2 -24 6 4 7 13 2 16 18 3 23 17 q - --- - q + --- - --- - --- + --- - --- - --- + --- + --- - --- + --- + 25 23 22 21 20 19 18 17 16 15 14 q q q q q q q q q q q 10 26 12 14 23 7 11 13 3 5 4 -2 1 > --- - --- + --- + --- - -- + -- + -- - -- + -- + -- - -- + q + - 13 12 11 10 9 8 7 6 5 4 3 q q q q q q q q q q q q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10₁₃₁

10₁₃₀
10₁₃₂

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-26 - 2q^-25 - q^-24 + 6q^-23 - 4q^-22 - 7q^-21 + 13q^-20 - 2q^-19 - 16q^-18 + 18q^-17 + 3q^-16 - 23q^-15 + 17q^-14 + 10q^-13 - 26q^-12 + 12q^-11 + 14q^-10 - 23q^-9 + 7q^-8 + 11q^-7 - 13q^-6 + 3q^-5 + 5q^-4 - 4q^-3 + q^-2 + q^-1
3	q^-51 - 2q^-50 - q^-49 + 2q^-48 + 5q^-47 - 3q^-46 - 9q^-45 + 15q^-43 + 5q^-42 - 18q^-41 - 15q^-40 + 19q^-39 + 25q^-38 - 15q^-37 - 33q^-36 + 6q^-35 + 41q^-34 + q^-33 - 40q^-32 - 16q^-31 + 43q^-30 + 24q^-29 - 37q^-28 - 38q^-27 + 35q^-26 + 46q^-25 - 27q^-24 - 57q^-23 + 22q^-22 + 63q^-21 - 15q^-20 - 63q^-19 + 4q^-18 + 63q^-17 - q^-16 - 50q^-15 - 9q^-14 + 43q^-13 + 6q^-12 - 25q^-11 - 10q^-10 + 18q^-9 + 3q^-8 - 6q^-7 - 5q^-6 + 7q^-5 - 2q^-4 + q^-3 - 2q^-2 + 2q^-1
4	q^-84 - 2q^-83 - q^-82 + 2q^-81 + q^-80 + 6q^-79 - 7q^-78 - 7q^-77 + 25q^-74 - 3q^-73 - 15q^-72 - 16q^-71 - 21q^-70 + 46q^-69 + 21q^-68 + 5q^-67 - 27q^-66 - 73q^-65 + 32q^-64 + 36q^-63 + 58q^-62 + 8q^-61 - 112q^-60 - 17q^-59 - 5q^-58 + 95q^-57 + 85q^-56 - 94q^-55 - 50q^-54 - 89q^-53 + 74q^-52 + 151q^-51 - 29q^-50 - 38q^-49 - 168q^-48 + 12q^-47 + 180q^-46 + 44q^-45 + q^-44 - 222q^-43 - 57q^-42 + 186q^-41 + 112q^-40 + 39q^-39 - 262q^-38 - 117q^-37 + 181q^-36 + 172q^-35 + 76q^-34 - 282q^-33 - 174q^-32 + 154q^-31 + 211q^-30 + 119q^-29 - 256q^-28 - 207q^-27 + 92q^-26 + 192q^-25 + 150q^-24 - 170q^-23 - 184q^-22 + 21q^-21 + 115q^-20 + 132q^-19 - 72q^-18 - 108q^-17 - 11q^-16 + 34q^-15 + 73q^-14 - 17q^-13 - 37q^-12 - 5q^-11 + 22q^-9 - 4q^-8 - 7q^-7 + 2q^-6 - 2q^-5 + 4q^-4 - q^-3 - 2q^-2 + q^-1 + 1

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[10, 131]]
Out[2]=	PD[X[1, 4, 2, 5], X[3, 8, 4, 9], X[14, 6, 15, 5], X[15, 20, 16, 1], > X[9, 16, 10, 17], X[19, 10, 20, 11], X[11, 18, 12, 19], X[17, 12, 18, 13], > X[6, 14, 7, 13], X[7, 2, 8, 3]]
In[3]:=	GaussCode[Knot[10, 131]]
Out[3]=	GaussCode[-1, 10, -2, 1, 3, -9, -10, 2, -5, 6, -7, 8, 9, -3, -4, 5, -8, 7, -6, > 4]
In[4]:=	DTCode[Knot[10, 131]]
Out[4]=	DTCode[4, 8, -14, 2, 16, 18, -6, 20, 12, 10]
In[5]:=	br = BR[Knot[10, 131]]
Out[5]=	BR[4, {-1, -1, -1, -2, 1, 1, -2, -2, -3, 2, -3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 11}
In[7]:=	BraidIndex[Knot[10, 131]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[10, 131]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 131]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 1, 2, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 131]][t]
Out[10]=	2 8 2 -11 - -- + - + 8 t - 2 t 2 t t
In[11]:=	Conway[Knot[10, 131]][z]
Out[11]=	4 1 - 2 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[8, 14], Knot[9, 8], Knot[10, 131]}
In[13]:=	{KnotDet[Knot[10, 131]], KnotSignature[Knot[10, 131]]}
Out[13]=	{31, -2}
In[14]:=	Jones[Knot[10, 131]][q]
Out[14]=	-9 2 3 5 5 5 5 3 2 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, 131]}
In[16]:=	A2Invariant[Knot[10, 131]][q]
Out[16]=	-28 -22 2 -18 -16 -14 -12 2 -6 2 q + q - --- - q - q - q + q + -- + q + -- 20 8 2 q q q
In[17]:=	HOMFLYPT[Knot[10, 131]][a, z]
Out[17]=	2 6 8 2 2 4 2 6 2 8 2 4 4 6 4 2 a - 2 a + a + 2 a z - a z - 2 a z + a z - a z - a z
In[18]:=	Kauffman[Knot[10, 131]][a, z]
Out[18]=	2 6 8 3 5 7 9 2 2 4 2 -2 a + 2 a + a + a z - a z - 5 a z - 3 a z + 3 a z + 2 a z - 6 2 8 2 10 2 3 3 5 3 7 3 9 3 > 3 a z + 2 a z + 4 a z + a z + 2 a z + 10 a z + 9 a z - 4 4 6 4 8 4 10 4 3 5 5 5 7 5 > 2 a z - 2 a z - 4 a z - 4 a z + a z - 3 a z - 12 a z - 9 5 4 6 8 6 10 6 5 7 7 7 9 7 6 8 > 8 a z + 2 a z - a z + a z + 2 a z + 4 a z + 2 a z + a z + 8 8 > a z
In[19]:=	{Vassiliev[2][Knot[10, 131]], Vassiliev[3][Knot[10, 131]]}
Out[19]=	{0, 2}
In[20]:=	Kh[Knot[10, 131]][q, t]
Out[20]=	-3 2 1 1 1 2 1 3 2 q + - + ------ + ------ + ------ + ------ + ------ + ------ + ------ + q 19 8 17 7 15 7 15 6 13 6 13 5 11 5 q t q t q t q t q t q t q t 2 3 3 2 2 3 1 2 > ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- 11 4 9 4 9 3 7 3 7 2 5 2 5 3 q t q t q t q t q t q t q t q t
In[21]:=	ColouredJones[Knot[10, 131], 2][q]
Out[21]=	-26 2 -24 6 4 7 13 2 16 18 3 23 17 q - --- - q + --- - --- - --- + --- - --- - --- + --- + --- - --- + --- + 25 23 22 21 20 19 18 17 16 15 14 q q q q q q q q q q q 10 26 12 14 23 7 11 13 3 5 4 -2 1 > --- - --- + --- + --- - -- + -- + -- - -- + -- + -- - -- + q + - 13 12 11 10 9 8 7 6 5 4 3 q q q q q q q q q q q q

The Non Alternating Knot 10131

The Non Alternating Knot 10₁₃₁