The Non Alternating Knot 10₁₄₀

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Acknowledgement

KnotPlot

PD Presentation: X₁₄₂₅ X_3,10,4,11 X_11,19,12,18 X_14,5,15,6 X_6,17,7,18 X_16,7,17,8 X_8,15,9,16 X_13,1,14,20 X_19,13,20,12 X_9,2,10,3

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

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

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 2

Alexander Polynomial: t^-2 - 2t^-1 + 3 - 2t + t²

Conway Polynomial: 1 + 2z² + z⁴

Other knots with the same Alexander/Conway Polynomial: {8₂₀, K11n73, K11n74, ...}

Determinant and Signature: {9, 0}

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

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

A2 (sl(3)) Invariant: - q^-22 - q^-20 - q^-18 + 2q^-14 + 2q^-12 + 2q^-10 - q^-6 - q^-4 + 1 + q²

HOMFLY-PT Polynomial: 1 - 2a² - a²z² + 4a⁴ + 4a⁴z² + a⁴z⁴ - 2a⁶ - a⁶z²

Kauffman Polynomial: 1 + 2a² - 4a²z² + a²z⁴ - 2a³z + 6a³z³ - 5a³z⁵ + a³z⁷ + 4a⁴ - 12a⁴z² + 12a⁴z⁴ - 6a⁴z⁶ + a⁴z⁸ - 6a⁵z + 16a⁵z³ - 11a⁵z⁵ + 2a⁵z⁷ + 2a⁶ - 8a⁶z² + 11a⁶z⁴ - 6a⁶z⁶ + a⁶z⁸ - 4a⁷z + 10a⁷z³ - 6a⁷z⁵ + a⁷z⁷

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

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 10₁₄₀. 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

j = 1 1

j = -1 1 1

j = -3

j = -5 1 1

j = -7 1

j = -9 1

j = -11 1 1

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 + 2q^-18 - q^-17 - 2q^-16 + 2q^-15 - q^-13 + 2q^-12 - q^-11 + q^-9 - 2q^-8 + q^-7 - 2q^-5 + 3q^-4 - 2q^-2 + 2q^-1 + 1 - q

3 - q^-42 + q^-41 + q^-40 - 2q^-38 + 2q^-36 + q^-35 - 2q^-34 - q^-33 + q^-32 - q^-30 + 2q^-28 - q^-27 - 2q^-26 + q^-25 + 4q^-24 - q^-23 - 4q^-22 + 5q^-20 - 5q^-18 - 2q^-17 + 5q^-16 + 2q^-15 - 5q^-14 - 3q^-13 + 6q^-12 + 3q^-11 - 5q^-10 - 4q^-9 + 6q^-8 + 4q^-7 - 5q^-6 - 5q^-5 + 5q^-4 + 4q^-3 - 2q^-2 - 4q^-1 + 2 + 2q - q³

4 q^-70 - q^-69 - q^-68 + 3q^-65 - q^-64 - q^-63 - q^-62 - 2q^-61 + 4q^-60 - 2q^-56 + 3q^-55 - 2q^-54 - q^-53 + 4q^-50 - q^-49 - 3q^-48 - 3q^-47 + 5q^-45 + 3q^-44 - 2q^-43 - 3q^-42 - 3q^-41 + 2q^-40 + 5q^-39 - 4q^-36 - 2q^-35 + 3q^-34 + 4q^-32 - 2q^-31 - 5q^-30 - q^-28 + 7q^-27 + q^-26 - 5q^-25 - 2q^-24 - 3q^-23 + 9q^-22 + 3q^-21 - 5q^-20 - 4q^-19 - 4q^-18 + 11q^-17 + 3q^-16 - 7q^-15 - 6q^-14 - 3q^-13 + 14q^-12 + 4q^-11 - 8q^-10 - 7q^-9 - 4q^-8 + 12q^-7 + 6q^-6 - 4q^-5 - 6q^-4 - 5q^-3 + 7q^-2 + 4q^-1 - 2q - 2q² + q³

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, 140]]

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

In[3]:=
GaussCode[Knot[10, 140]]

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

In[4]:=
DTCode[Knot[10, 140]]

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

In[5]:=
br = BR[Knot[10, 140]]

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

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

Out[6]=
{4, 11}

In[7]:=
BraidIndex[Knot[10, 140]]

Out[7]=
4

In[8]:=
Show[DrawMorseLink[Knot[10, 140]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[10, 140]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

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

In[10]:=
alex = Alexander[Knot[10, 140]][t]

Out[10]=
-2 2 2 3 + t - - - 2 t + t t

In[11]:=
Conway[Knot[10, 140]][z]

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

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

Out[12]=
{Knot[8, 20], Knot[10, 140], Knot[11, NonAlternating, 73], > Knot[11, NonAlternating, 74]}

In[13]:=
{KnotDet[Knot[10, 140]], KnotSignature[Knot[10, 140]]}

Out[13]=
{9, 0}

In[14]:=
Jones[Knot[10, 140]][q]

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

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

Out[15]=
{Knot[10, 140]}

In[16]:=
A2Invariant[Knot[10, 140]][q]

Out[16]=
-22 -20 -18 2 2 2 -6 -4 2 1 - q - q - q + --- + --- + --- - q - q + q 14 12 10 q q q

In[17]:=
HOMFLYPT[Knot[10, 140]][a, z]

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

In[18]:=
Kauffman[Knot[10, 140]][a, z]

Out[18]=
2 4 6 3 5 7 2 2 4 2 1 + 2 a + 4 a + 2 a - 2 a z - 6 a z - 4 a z - 4 a z - 12 a z - 6 2 3 3 5 3 7 3 2 4 4 4 6 4 > 8 a z + 6 a z + 16 a z + 10 a z + a z + 12 a z + 11 a z - 3 5 5 5 7 5 4 6 6 6 3 7 5 7 > 5 a z - 11 a z - 6 a z - 6 a z - 6 a z + a z + 2 a z + 7 7 4 8 6 8 > a z + a z + a z

In[19]:=
{Vassiliev[2][Knot[10, 140]], Vassiliev[3][Knot[10, 140]]}

Out[19]=
{2, -4}

In[20]:=
Kh[Knot[10, 140]][q, t]

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

In[21]:=
ColouredJones[Knot[10, 140], 2][q]

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

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10₁₄₀

10₁₃₉
10₁₄₁

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-21 - q^-20 - q^-19 + 2q^-18 - q^-17 - 2q^-16 + 2q^-15 - q^-13 + 2q^-12 - q^-11 + q^-9 - 2q^-8 + q^-7 - 2q^-5 + 3q^-4 - 2q^-2 + 2q^-1 + 1 - q
3	- q^-42 + q^-41 + q^-40 - 2q^-38 + 2q^-36 + q^-35 - 2q^-34 - q^-33 + q^-32 - q^-30 + 2q^-28 - q^-27 - 2q^-26 + q^-25 + 4q^-24 - q^-23 - 4q^-22 + 5q^-20 - 5q^-18 - 2q^-17 + 5q^-16 + 2q^-15 - 5q^-14 - 3q^-13 + 6q^-12 + 3q^-11 - 5q^-10 - 4q^-9 + 6q^-8 + 4q^-7 - 5q^-6 - 5q^-5 + 5q^-4 + 4q^-3 - 2q^-2 - 4q^-1 + 2 + 2q - q³
4	q^-70 - q^-69 - q^-68 + 3q^-65 - q^-64 - q^-63 - q^-62 - 2q^-61 + 4q^-60 - 2q^-56 + 3q^-55 - 2q^-54 - q^-53 + 4q^-50 - q^-49 - 3q^-48 - 3q^-47 + 5q^-45 + 3q^-44 - 2q^-43 - 3q^-42 - 3q^-41 + 2q^-40 + 5q^-39 - 4q^-36 - 2q^-35 + 3q^-34 + 4q^-32 - 2q^-31 - 5q^-30 - q^-28 + 7q^-27 + q^-26 - 5q^-25 - 2q^-24 - 3q^-23 + 9q^-22 + 3q^-21 - 5q^-20 - 4q^-19 - 4q^-18 + 11q^-17 + 3q^-16 - 7q^-15 - 6q^-14 - 3q^-13 + 14q^-12 + 4q^-11 - 8q^-10 - 7q^-9 - 4q^-8 + 12q^-7 + 6q^-6 - 4q^-5 - 6q^-4 - 5q^-3 + 7q^-2 + 4q^-1 - 2q - 2q² + q³

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[10, 140]]
Out[2]=	PD[X[1, 4, 2, 5], X[3, 10, 4, 11], X[11, 19, 12, 18], X[14, 5, 15, 6], > X[6, 17, 7, 18], X[16, 7, 17, 8], X[8, 15, 9, 16], X[13, 1, 14, 20], > X[19, 13, 20, 12], X[9, 2, 10, 3]]
In[3]:=	GaussCode[Knot[10, 140]]
Out[3]=	GaussCode[-1, 10, -2, 1, 4, -5, 6, -7, -10, 2, -3, 9, -8, -4, 7, -6, 5, 3, -9, > 8]
In[4]:=	DTCode[Knot[10, 140]]
Out[4]=	DTCode[4, 10, -14, -16, 2, 18, 20, -8, -6, 12]
In[5]:=	br = BR[Knot[10, 140]]
Out[5]=	BR[4, {1, 1, 1, -2, -1, -1, -1, -2, -3, 2, -3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 11}
In[7]:=	BraidIndex[Knot[10, 140]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[10, 140]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 140]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 2, 3, NotAvailable, 2}
In[10]:=	alex = Alexander[Knot[10, 140]][t]
Out[10]=	-2 2 2 3 + t - - - 2 t + t t
In[11]:=	Conway[Knot[10, 140]][z]
Out[11]=	2 4 1 + 2 z + z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[8, 20], Knot[10, 140], Knot[11, NonAlternating, 73], > Knot[11, NonAlternating, 74]}
In[13]:=	{KnotDet[Knot[10, 140]], KnotSignature[Knot[10, 140]]}
Out[13]=	{9, 0}
In[14]:=	Jones[Knot[10, 140]][q]
Out[14]=	-7 -6 -5 2 -3 -2 1 1 - q + q - q + -- - q + q - - 4 q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 140]}
In[16]:=	A2Invariant[Knot[10, 140]][q]
Out[16]=	-22 -20 -18 2 2 2 -6 -4 2 1 - q - q - q + --- + --- + --- - q - q + q 14 12 10 q q q
In[17]:=	HOMFLYPT[Knot[10, 140]][a, z]
Out[17]=	2 4 6 2 2 4 2 6 2 4 4 1 - 2 a + 4 a - 2 a - a z + 4 a z - a z + a z
In[18]:=	Kauffman[Knot[10, 140]][a, z]
Out[18]=	2 4 6 3 5 7 2 2 4 2 1 + 2 a + 4 a + 2 a - 2 a z - 6 a z - 4 a z - 4 a z - 12 a z - 6 2 3 3 5 3 7 3 2 4 4 4 6 4 > 8 a z + 6 a z + 16 a z + 10 a z + a z + 12 a z + 11 a z - 3 5 5 5 7 5 4 6 6 6 3 7 5 7 > 5 a z - 11 a z - 6 a z - 6 a z - 6 a z + a z + 2 a z + 7 7 4 8 6 8 > a z + a z + a z
In[19]:=	{Vassiliev[2][Knot[10, 140]], Vassiliev[3][Knot[10, 140]]}
Out[19]=	{2, -4}
In[20]:=	Kh[Knot[10, 140]][q, t]
Out[20]=	1 1 1 1 1 1 1 1 1 - + q + ------ + ------ + ------ + ----- + ----- + ----- + ----- + --- q 15 7 11 6 11 5 9 4 7 4 5 3 5 2 q t q t q t q t q t q t q t q t
In[21]:=	ColouredJones[Knot[10, 140], 2][q]
Out[21]=	-21 -20 -19 2 -17 2 2 -13 2 -11 -9 1 + q - q - q + --- - q - --- + --- - q + --- - q + q - 18 16 15 12 q q q q 2 -7 2 3 2 2 > -- + q - -- + -- - -- + - - q 8 5 4 2 q q q q q

The Non Alternating Knot 10140

The Non Alternating Knot 10₁₄₀