The Non Alternating Knot 10₁₃₀

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Acknowledgement

KnotPlot

PD Presentation: X₄₂₅₁ X₈₄₉₃ X_5,14,6,15 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_13,6,14,7 X₂₈₃₇

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 2 2 3 / NotAvailable 1

Alexander Polynomial: 2t^-2 - 4t^-1 + 5 - 4t + 2t²

Conway Polynomial: 1 + 4z² + 2z⁴

Other knots with the same Alexander/Conway Polynomial: {7₅, ...}

Determinant and Signature: {17, 0}

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

Other knots (up to mirrors) with the same Jones Polynomial: {K11n61, ...}

A2 (sl(3)) Invariant: - q^-22 - q^-20 - q^-18 - q^-16 + q^-14 + q^-12 + 2q^-10 + 2q^-8 + q^-6 + q^-4 - q⁴

HOMFLY-PT Polynomial: - 1 - z² + 2a² + 3a²z² + a²z⁴ + 2a⁴ + 3a⁴z² + a⁴z⁴ - 2a⁶ - a⁶z²

Kauffman Polynomial: a^-1z - 1 + 2z² + az - 2az³ + az⁵ - 2a² + 6a²z² - 7a²z⁴ + 2a²z⁶ - 3a³z + 8a³z³ - 8a³z⁵ + 2a³z⁷ + 2a⁴ - 3a⁴z⁶ + a⁴z⁸ - 9a⁵z + 21a⁵z³ - 15a⁵z⁵ + 3a⁵z⁷ + 2a⁶ - 4a⁶z² + 7a⁶z⁴ - 5a⁶z⁶ + a⁶z⁸ - 6a⁷z + 11a⁷z³ - 6a⁷z⁵ + a⁷z⁷

V₂ and V₃, the type 2 and 3 Vassiliev invariants: {4, -6}

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=0 is the signature of 10₁₃₀. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

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

j = 3 1

j = 1 1

j = -1 2 2

j = -3 1

j = -5 1 2

j = -7 2 1

j = -9 1

j = -11 1 2

j = -13

j = -15 1

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

3 - q^-42 + q^-41 + q^-40 - 3q^-38 + 3q^-36 + 3q^-35 - 4q^-34 - 3q^-33 + 2q^-32 + 5q^-31 - 2q^-30 - 3q^-29 + 2q^-27 - q^-26 + 2q^-24 - 3q^-23 - 3q^-22 + 3q^-21 + 6q^-20 - 6q^-19 - 6q^-18 + 4q^-17 + 10q^-16 - 7q^-15 - 9q^-14 + 5q^-13 + 15q^-12 - 9q^-11 - 13q^-10 + 6q^-9 + 18q^-8 - 8q^-7 - 15q^-6 + 4q^-5 + 17q^-4 - 4q^-3 - 12q^-2 + q^-1 + 9 - q - 5q² + q³ + 2q⁴ - 2q⁶ + q⁷

4 q^-70 - q^-69 - q^-68 + 4q^-65 - q^-64 - 2q^-63 - 2q^-62 - 4q^-61 + 7q^-60 + 2q^-59 + q^-58 - 2q^-57 - 9q^-56 + 5q^-55 + 4q^-53 + 3q^-52 - 7q^-51 + 5q^-50 - 5q^-49 - 2q^-48 + 3q^-47 - 2q^-46 + 14q^-45 - 3q^-44 - 11q^-43 - 5q^-42 - 5q^-41 + 22q^-40 + 9q^-39 - 13q^-38 - 12q^-37 - 15q^-36 + 20q^-35 + 21q^-34 - 7q^-33 - 11q^-32 - 24q^-31 + 10q^-30 + 27q^-29 - 5q^-27 - 29q^-26 + 29q^-24 + 5q^-23 + q^-22 - 33q^-21 - 8q^-20 + 32q^-19 + 11q^-18 + 4q^-17 - 38q^-16 - 17q^-15 + 35q^-14 + 19q^-13 + 11q^-12 - 41q^-11 - 27q^-10 + 30q^-9 + 22q^-8 + 18q^-7 - 31q^-6 - 28q^-5 + 16q^-4 + 13q^-3 + 19q^-2 - 15q^-1 - 17 + 7q + q² + 10q³ - 5q⁴ - 5q⁵ + 4q⁶ - 2q⁷ + 2q⁸ - 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, 130]]

Out[2]=
PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[5, 14, 6, 15], 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[13, 6, 14, 7], X[2, 8, 3, 7]]

In[3]:=
GaussCode[Knot[10, 130]]

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, 130]]

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

In[5]:=
br = BR[Knot[10, 130]]

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, 130]]

Out[7]=
4

In[8]:=
Show[DrawMorseLink[Knot[10, 130]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[10, 130]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

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

In[10]:=
alex = Alexander[Knot[10, 130]][t]

Out[10]=
2 4 2 5 + -- - - - 4 t + 2 t 2 t t

In[11]:=
Conway[Knot[10, 130]][z]

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

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

Out[12]=
{Knot[7, 5], Knot[10, 130]}

In[13]:=
{KnotDet[Knot[10, 130]], KnotSignature[Knot[10, 130]]}

Out[13]=
{17, 0}

In[14]:=
Jones[Knot[10, 130]][q]

Out[14]=
-7 -6 2 3 2 3 2 2 - q + q - -- + -- - -- + -- - - - q 5 4 3 2 q q q q q

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

Out[15]=
{Knot[10, 130], Knot[11, NonAlternating, 61]}

In[16]:=
A2Invariant[Knot[10, 130]][q]

Out[16]=
-22 -20 -18 -16 -14 -12 2 2 -6 -4 4 -q - q - q - q + q + q + --- + -- + q + q - q 10 8 q q

In[17]:=
HOMFLYPT[Knot[10, 130]][a, z]

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

In[18]:=
Kauffman[Knot[10, 130]][a, z]

Out[18]=
2 4 6 z 3 5 7 2 2 2 -1 - 2 a + 2 a + 2 a + - + a z - 3 a z - 9 a z - 6 a z + 2 z + 6 a z - a 6 2 3 3 3 5 3 7 3 2 4 6 4 > 4 a z - 2 a z + 8 a z + 21 a z + 11 a z - 7 a z + 7 a z + 5 3 5 5 5 7 5 2 6 4 6 6 6 > a z - 8 a z - 15 a z - 6 a z + 2 a z - 3 a z - 5 a z + 3 7 5 7 7 7 4 8 6 8 > 2 a z + 3 a z + a z + a z + a z

In[19]:=
{Vassiliev[2][Knot[10, 130]], Vassiliev[3][Knot[10, 130]]}

Out[19]=
{4, -6}

In[20]:=
Kh[Knot[10, 130]][q, t]

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

In[21]:=
ColouredJones[Knot[10, 130], 2][q]

Out[21]=
-21 -20 -19 3 -17 4 4 5 4 -11 4 -1 + q - q - q + --- - q - --- + --- - --- + --- + q - --- + 18 16 15 13 12 10 q q q q q q 2 2 2 2 -4 2 -2 1 > -- + -- - -- + -- + q - -- + q + - 9 8 7 5 3 q q q q q q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10₁₃₀

10₁₂₉
10₁₃₁

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-21 - q^-20 - q^-19 + 3q^-18 - q^-17 - 4q^-16 + 4q^-15 - 5q^-13 + 4q^-12 + q^-11 - 4q^-10 + 2q^-9 + 2q^-8 - 2q^-7 + 2q^-5 + q^-4 - 2q^-3 + q^-2 + q^-1 - 1
3	- q^-42 + q^-41 + q^-40 - 3q^-38 + 3q^-36 + 3q^-35 - 4q^-34 - 3q^-33 + 2q^-32 + 5q^-31 - 2q^-30 - 3q^-29 + 2q^-27 - q^-26 + 2q^-24 - 3q^-23 - 3q^-22 + 3q^-21 + 6q^-20 - 6q^-19 - 6q^-18 + 4q^-17 + 10q^-16 - 7q^-15 - 9q^-14 + 5q^-13 + 15q^-12 - 9q^-11 - 13q^-10 + 6q^-9 + 18q^-8 - 8q^-7 - 15q^-6 + 4q^-5 + 17q^-4 - 4q^-3 - 12q^-2 + q^-1 + 9 - q - 5q² + q³ + 2q⁴ - 2q⁶ + q⁷
4	q^-70 - q^-69 - q^-68 + 4q^-65 - q^-64 - 2q^-63 - 2q^-62 - 4q^-61 + 7q^-60 + 2q^-59 + q^-58 - 2q^-57 - 9q^-56 + 5q^-55 + 4q^-53 + 3q^-52 - 7q^-51 + 5q^-50 - 5q^-49 - 2q^-48 + 3q^-47 - 2q^-46 + 14q^-45 - 3q^-44 - 11q^-43 - 5q^-42 - 5q^-41 + 22q^-40 + 9q^-39 - 13q^-38 - 12q^-37 - 15q^-36 + 20q^-35 + 21q^-34 - 7q^-33 - 11q^-32 - 24q^-31 + 10q^-30 + 27q^-29 - 5q^-27 - 29q^-26 + 29q^-24 + 5q^-23 + q^-22 - 33q^-21 - 8q^-20 + 32q^-19 + 11q^-18 + 4q^-17 - 38q^-16 - 17q^-15 + 35q^-14 + 19q^-13 + 11q^-12 - 41q^-11 - 27q^-10 + 30q^-9 + 22q^-8 + 18q^-7 - 31q^-6 - 28q^-5 + 16q^-4 + 13q^-3 + 19q^-2 - 15q^-1 - 17 + 7q + q² + 10q³ - 5q⁴ - 5q⁵ + 4q⁶ - 2q⁷ + 2q⁸ - 2q⁹ + 2q¹¹ - q¹²

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[10, 130]]
Out[2]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[5, 14, 6, 15], 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[13, 6, 14, 7], X[2, 8, 3, 7]]
In[3]:=	GaussCode[Knot[10, 130]]
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, 130]]
Out[4]=	DTCode[4, 8, -14, 2, -16, -18, -6, -20, -12, -10]
In[5]:=	br = BR[Knot[10, 130]]
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, 130]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[10, 130]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 130]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 2, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 130]][t]
Out[10]=	2 4 2 5 + -- - - - 4 t + 2 t 2 t t
In[11]:=	Conway[Knot[10, 130]][z]
Out[11]=	2 4 1 + 4 z + 2 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[7, 5], Knot[10, 130]}
In[13]:=	{KnotDet[Knot[10, 130]], KnotSignature[Knot[10, 130]]}
Out[13]=	{17, 0}
In[14]:=	Jones[Knot[10, 130]][q]
Out[14]=	-7 -6 2 3 2 3 2 2 - q + q - -- + -- - -- + -- - - - q 5 4 3 2 q q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 130], Knot[11, NonAlternating, 61]}
In[16]:=	A2Invariant[Knot[10, 130]][q]
Out[16]=	-22 -20 -18 -16 -14 -12 2 2 -6 -4 4 -q - q - q - q + q + q + --- + -- + q + q - q 10 8 q q
In[17]:=	HOMFLYPT[Knot[10, 130]][a, z]
Out[17]=	2 4 6 2 2 2 4 2 6 2 2 4 4 4 -1 + 2 a + 2 a - 2 a - z + 3 a z + 3 a z - a z + a z + a z
In[18]:=	Kauffman[Knot[10, 130]][a, z]
Out[18]=	2 4 6 z 3 5 7 2 2 2 -1 - 2 a + 2 a + 2 a + - + a z - 3 a z - 9 a z - 6 a z + 2 z + 6 a z - a 6 2 3 3 3 5 3 7 3 2 4 6 4 > 4 a z - 2 a z + 8 a z + 21 a z + 11 a z - 7 a z + 7 a z + 5 3 5 5 5 7 5 2 6 4 6 6 6 > a z - 8 a z - 15 a z - 6 a z + 2 a z - 3 a z - 5 a z + 3 7 5 7 7 7 4 8 6 8 > 2 a z + 3 a z + a z + a z + a z
In[19]:=	{Vassiliev[2][Knot[10, 130]], Vassiliev[3][Knot[10, 130]]}
Out[19]=	{4, -6}
In[20]:=	Kh[Knot[10, 130]][q, t]
Out[20]=	2 1 1 2 1 2 1 1 2 - + q + ------ + ------ + ------ + ----- + ----- + ----- + ----- + ----- + q 15 7 11 6 11 5 9 4 7 4 7 3 5 3 5 2 q t q t q t q t q t q t q t q t 1 2 3 > ----- + --- + q t 3 2 q t q t
In[21]:=	ColouredJones[Knot[10, 130], 2][q]
Out[21]=	-21 -20 -19 3 -17 4 4 5 4 -11 4 -1 + q - q - q + --- - q - --- + --- - --- + --- + q - --- + 18 16 15 13 12 10 q q q q q q 2 2 2 2 -4 2 -2 1 > -- + -- - -- + -- + q - -- + q + - 9 8 7 5 3 q q q q q q

The Non Alternating Knot 10130

The Non Alternating Knot 10₁₃₀