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

K11a117 K11a119

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 14 18 12 2 8 20 22 6 16

Alexander Polynomial:

2t-3 - 10t-2 + 20t-1 - 23 + 20t - 10t2 + 2t3

Conway Polynomial:

1 - 2z2 + 2z4 + 2z6

Other knots with the same Alexander/Conway Polynomial:

{K11a90, ...}

Determinant and Signature:

{87, 2}

Jones Polynomial:

q-3 - 2q-2 + 5q-1 - 8 + 11q - 14q2 + 14q3 - 12q4 + 10q5 - 6q6 + 3q7 - q8

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

{K11a45, ...}

A2 (sl(3)) Invariant:

q-10 + q-8 + 2q-4 - q-2 - 1 + q2 - 4q4 + q6 - q8 + q10 + 3q12 - q14 + 3q16 - q18 - q20 + q22 - q24

HOMFLY-PT Polynomial:

- a-6 - 2a-6z2 - a-6z4 + 3a-4 + 4a-4z2 + 3a-4z4 + a-4z6 - a-2 + 2a-2z4 + a-2z6 - 2 - 5z2 - 2z4 + 2a2 + a2z2

Kauffman Polynomial:

- 2a-9z3 + a-9z5 + a-8z2 - 6a-8z4 + 3a-8z6 - a-7z + 6a-7z3 - 11a-7z5 + 5a-7z7 + a-6 - 10a-6z2 + 21a-6z4 - 17a-6z6 + 6a-6z8 + a-5z - 2a-5z3 + 10a-5z5 - 9a-5z7 + 4a-5z9 + 3a-4 - 24a-4z2 + 43a-4z4 - 26a-4z6 + 6a-4z8 + a-4z10 + 7a-3z - 23a-3z3 + 31a-3z5 - 19a-3z7 + 6a-3z9 + a-2 - 10a-2z2 + 13a-2z4 - 9a-2z6 + 2a-2z8 + a-2z10 + 5a-1z - 9a-1z3 + 3a-1z5 - 3a-1z7 + 2a-1z9 - 2 + 8z2 - 7z4 - 2z6 + 2z8 + 4az3 - 6az5 + 2az7 - 2a2 + 5a2z2 - 4a2z4 + a2z6

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

{-2, 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 11118. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 17                       1 j = 15                     2   j = 13                   4 1   j = 11                 6 2     j = 9               6 4       j = 7             8 6         j = 5           6 6           j = 3         5 8             j = 1       4 7               j = -1     1 4                 j = -3   1 4                   j = -5   1                     j = -7 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, 118]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 118]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[14, 6, 15, 5], X[18, 7, 19, 8], > X[12, 10, 13, 9], X[2, 11, 3, 12], X[8, 14, 9, 13], X[20, 16, 21, 15], > X[22, 18, 1, 17], X[6, 19, 7, 20], X[16, 22, 17, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 118]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -10, 4, -7, 5, -2, 6, -5, 7, -3, 8, -11, 9, -4, 10, > -8, 11, -9]
In[5]:=	DTCode[Knot[11, Alternating, 118]]
Out[5]=	DTCode[4, 10, 14, 18, 12, 2, 8, 20, 22, 6, 16]
In[6]:=	alex = Alexander[Knot[11, Alternating, 118]][t]
Out[6]=	2 10 20 2 3 -23 + -- - -- + -- + 20 t - 10 t + 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 118]][z]
Out[7]=	2 4 6 1 - 2 z + 2 z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 90], Knot[11, Alternating, 118]}
In[9]:=	{KnotDet[Knot[11, Alternating, 118]], KnotSignature[Knot[11, Alternating, 118]]}
Out[9]=	{87, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 118]][q]
Out[10]=	-3 2 5 2 3 4 5 6 7 8 -8 + q - -- + - + 11 q - 14 q + 14 q - 12 q + 10 q - 6 q + 3 q - q 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 45], Knot[11, Alternating, 118]}
In[12]:=	A2Invariant[Knot[11, Alternating, 118]][q]
Out[12]=	-10 -8 2 -2 2 4 6 8 10 12 14 16 -1 + q + q + -- - q + q - 4 q + q - q + q + 3 q - q + 3 q - 4 q 18 20 22 24 > q - q + q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 118]][a, z]
Out[13]=	2 2 4 4 -6 3 -2 2 2 2 z 4 z 2 2 4 z 3 z -2 - a + -- - a + 2 a - 5 z - ---- + ---- + a z - 2 z - -- + ---- + 4 6 4 6 4 a a a a a 4 6 6 2 z z z > ---- + -- + -- 2 4 2 a a a
In[14]:=	Kauffman[Knot[11, Alternating, 118]][a, z]
Out[14]=	2 2 2 -6 3 -2 2 z z 7 z 5 z 2 z 10 z 24 z -2 + a + -- + a - 2 a - -- + -- + --- + --- + 8 z + -- - ----- - ----- - 4 7 5 3 a 8 6 4 a a a a a a a 2 3 3 3 3 3 10 z 2 2 2 z 6 z 2 z 23 z 9 z 3 4 > ----- + 5 a z - ---- + ---- - ---- - ----- - ---- + 4 a z - 7 z - 2 9 7 5 3 a a a a a a 4 4 4 4 5 5 5 5 6 z 21 z 43 z 13 z 2 4 z 11 z 10 z 31 z > ---- + ----- + ----- + ----- - 4 a z + -- - ----- + ----- + ----- + 8 6 4 2 9 7 5 3 a a a a a a a a 5 6 6 6 6 7 7 3 z 5 6 3 z 17 z 26 z 9 z 2 6 5 z 9 z > ---- - 6 a z - 2 z + ---- - ----- - ----- - ---- + a z + ---- - ---- - a 8 6 4 2 7 5 a a a a a a 7 7 8 8 8 9 9 9 19 z 3 z 7 8 6 z 6 z 2 z 4 z 6 z 2 z > ----- - ---- + 2 a z + 2 z + ---- + ---- + ---- + ---- + ---- + ---- + 3 a 6 4 2 5 3 a a a a a a a 10 10 z z > --- + --- 4 2 a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 118]], Vassiliev[3][Knot[11, Alternating, 118]]}
Out[15]=	{-2, 1}
In[16]:=	Kh[Knot[11, Alternating, 118]][q, t]
Out[16]=	3 1 1 1 4 1 4 4 q 3 7 q + 5 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 8 q t + 7 4 5 3 3 3 3 2 2 q t t q t q t q t q t q t 5 5 2 7 2 7 3 9 3 9 4 11 4 > 6 q t + 6 q t + 8 q t + 6 q t + 6 q t + 4 q t + 6 q t + 11 5 13 5 13 6 15 6 17 7 > 2 q t + 4 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a118

K11a117 K11a119