© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:

K11a159 K11a161

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X10,4,11,3 X18,5,19,6 X14,7,15,8 X2,10,3,9 X22,11,1,12 X20,14,21,13 X8,15,9,16 X12,18,13,17 X6,19,7,20 X16,22,17,21

Gauss Code:

{1, -5, 2, -1, 3, -10, 4, -8, 5, -2, 6, -9, 7, -4, 8, -11, 9, -3, 10, -7, 11, -6}

DT (Dowker-Thistlethwaite) Code:

4 10 18 14 2 22 20 8 12 6 16

Alexander Polynomial:

t-4 - 6t-3 + 17t-2 - 30t-1 + 37 - 30t + 17t2 - 6t3 + t4

Conway Polynomial:

1 + z4 + 2z6 + z8

Other knots with the same Alexander/Conway Polynomial:

{K11a76, K11a289, ...}

Determinant and Signature:

{145, 0}

Jones Polynomial:

- q-5 + 4q-4 - 9q-3 + 15q-2 - 20q-1 + 24 - 23q + 20q2 - 15q3 + 9q4 - 4q5 + q6

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

{K11a76, K11a289, ...}

A2 (sl(3)) Invariant:

- q-14 + 2q-12 - 3q-10 + 2q-8 + q-6 - 3q-4 + 6q-2 - 3 + 4q2 - q4 - 2q6 + 3q8 - 4q10 + 2q12 - q16 + q18

HOMFLY-PT Polynomial:

a-4 + 2a-4z2 + a-4z4 - 3a-2 - 8a-2z2 - 7a-2z4 - 2a-2z6 + 4 + 9z2 + 10z4 + 5z6 + z8 - a2 - 3a2z2 - 3a2z4 - a2z6

Kauffman Polynomial:

a-6z2 - 2a-6z4 + a-6z6 - 2a-5z + 6a-5z3 - 9a-5z5 + 4a-5z7 + a-4 - 4a-4z2 + 9a-4z4 - 15a-4z6 + 7a-4z8 - 6a-3z + 18a-3z3 - 16a-3z5 - 4a-3z7 + 6a-3z9 + 3a-2 - 18a-2z2 + 44a-2z4 - 47a-2z6 + 15a-2z8 + 2a-2z10 - 8a-1z + 24a-1z3 - 11a-1z5 - 17a-1z7 + 13a-1z9 + 4 - 20z2 + 50z4 - 51z6 + 18z8 + 2z10 - 6az + 19az3 - 17az5 - az7 + 7az9 + a2 - 6a2z2 + 12a2z4 - 16a2z6 + 10a2z8 - 2a3z + 6a3z3 - 12a3z5 + 8a3z7 + a4z2 - 5a4z4 + 4a4z6 - a5z3 + a5z5

V2 and V3, the type 2 and 3 Vassiliev invariants:

{0, -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=0 is the signature of 11160. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 13                       1 j = 11                     3   j = 9                   6 1   j = 7                 9 3     j = 5               11 6       j = 3             12 9         j = 1           12 11           j = -1         9 13             j = -3       6 11               j = -5     3 9                 j = -7   1 6                   j = -9   3                     j = -11 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]:=	Show[DrawMorseLink[Knot[11, Alternating, 160]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 160]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[18, 5, 19, 6], X[14, 7, 15, 8], > X[2, 10, 3, 9], X[22, 11, 1, 12], X[20, 14, 21, 13], X[8, 15, 9, 16], > X[12, 18, 13, 17], X[6, 19, 7, 20], X[16, 22, 17, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 160]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -10, 4, -8, 5, -2, 6, -9, 7, -4, 8, -11, 9, -3, 10, > -7, 11, -6]
In[5]:=	DTCode[Knot[11, Alternating, 160]]
Out[5]=	DTCode[4, 10, 18, 14, 2, 22, 20, 8, 12, 6, 16]
In[6]:=	alex = Alexander[Knot[11, Alternating, 160]][t]
Out[6]=	-4 6 17 30 2 3 4 37 + t - -- + -- - -- - 30 t + 17 t - 6 t + t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 160]][z]
Out[7]=	4 6 8 1 + z + 2 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 76], Knot[11, Alternating, 160], > Knot[11, Alternating, 289]}
In[9]:=	{KnotDet[Knot[11, Alternating, 160]], KnotSignature[Knot[11, Alternating, 160]]}
Out[9]=	{145, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 160]][q]
Out[10]=	-5 4 9 15 20 2 3 4 5 6 24 - q + -- - -- + -- - -- - 23 q + 20 q - 15 q + 9 q - 4 q + q 4 3 2 q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 76], Knot[11, Alternating, 160], > Knot[11, Alternating, 289]}
In[12]:=	A2Invariant[Knot[11, Alternating, 160]][q]
Out[12]=	-14 2 3 2 -6 3 6 2 4 6 8 10 -3 - q + --- - --- + -- + q - -- + -- + 4 q - q - 2 q + 3 q - 4 q + 12 10 8 4 2 q q q q q 12 16 18 > 2 q - q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 160]][a, z]
Out[13]=	2 2 4 4 -4 3 2 2 2 z 8 z 2 2 4 z 7 z 4 + a - -- - a + 9 z + ---- - ---- - 3 a z + 10 z + -- - ---- - 2 4 2 4 2 a a a a a 6 2 4 6 2 z 2 6 8 > 3 a z + 5 z - ---- - a z + z 2 a
In[14]:=	Kauffman[Knot[11, Alternating, 160]][a, z]
Out[14]=	2 2 -4 3 2 2 z 6 z 8 z 3 2 z 4 z 4 + a + -- + a - --- - --- - --- - 6 a z - 2 a z - 20 z + -- - ---- - 2 5 3 a 6 4 a a a a a 2 3 3 3 18 z 2 2 4 2 6 z 18 z 24 z 3 3 3 > ----- - 6 a z + a z + ---- + ----- + ----- + 19 a z + 6 a z - 2 5 3 a a a a 4 4 4 5 5 5 3 4 2 z 9 z 44 z 2 4 4 4 9 z 16 z > a z + 50 z - ---- + ---- + ----- + 12 a z - 5 a z - ---- - ----- - 6 4 2 5 3 a a a a a 5 6 6 6 11 z 5 3 5 5 5 6 z 15 z 47 z > ----- - 17 a z - 12 a z + a z - 51 z + -- - ----- - ----- - a 6 4 2 a a a 7 7 7 8 2 6 4 6 4 z 4 z 17 z 7 3 7 8 7 z > 16 a z + 4 a z + ---- - ---- - ----- - a z + 8 a z + 18 z + ---- + 5 3 a 4 a a a 8 9 9 10 15 z 2 8 6 z 13 z 9 10 2 z > ----- + 10 a z + ---- + ----- + 7 a z + 2 z + ----- 2 3 a 2 a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 160]], Vassiliev[3][Knot[11, Alternating, 160]]}
Out[15]=	{0, -1}
In[16]:=	Kh[Knot[11, Alternating, 160]][q, t]
Out[16]=	13 1 3 1 6 3 9 6 11 -- + 12 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ---- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t q t 9 3 3 2 5 2 5 3 7 3 7 4 > --- + 11 q t + 12 q t + 9 q t + 11 q t + 6 q t + 9 q t + 3 q t + q t 9 4 9 5 11 5 13 6 > 6 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a160

K11a159 K11a161