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

K11a77 K11a79

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 12 14 20 2 16 6 22 8 18

Alexander Polynomial:

2t-3 - 12t-2 + 29t-1 - 37 + 29t - 12t2 + 2t3

Conway Polynomial:

1 - z2 + 2z6

Other knots with the same Alexander/Conway Polynomial:

{K11a130, ...}

Determinant and Signature:

{123, 2}

Jones Polynomial:

q-3 - 3q-2 + 7q-1 - 12 + 17q - 19q2 + 20q3 - 18q4 + 13q5 - 8q6 + 4q7 - q8

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

{...}

A2 (sl(3)) Invariant:

q-10 - q-6 + 3q-4 - 2q-2 - 1 + 3q2 - 3q4 + 4q6 - q8 + q12 - 4q14 + 4q16 - q18 - q20 + 2q22 - q24

HOMFLY-PT Polynomial:

- a-6z2 - a-6z4 + 2a-4z2 + 2a-4z4 + a-4z6 + a-2 + a-2z4 + a-2z6 - 1 - 3z2 - 2z4 + a2 + a2z2

Kauffman Polynomial:

- a-9z3 + a-9z5 + 2a-8z2 - 6a-8z4 + 4a-8z6 - a-7z + 6a-7z3 - 11a-7z5 + 7a-7z7 + 3a-6z2 - 3a-6z4 - 6a-6z6 + 7a-6z8 - 4a-5z + 19a-5z3 - 24a-5z5 + 7a-5z7 + 4a-5z9 - a-4z2 + 11a-4z4 - 24a-4z6 + 13a-4z8 + a-4z10 - 6a-3z + 22a-3z3 - 23a-3z5 + 7a-3z9 - a-2 + 9a-2z4 - 21a-2z6 + 10a-2z8 + a-2z10 - 5a-1z + 17a-1z3 - 19a-1z5 + 3a-1z7 + 3a-1z9 - 1 + 5z2 - 2z4 - 6z6 + 4z8 - 2az + 7az3 - 8az5 + 3az7 - a2 + 3a2z2 - 3a2z4 + a2z6

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

{-1, 0}

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 1178. 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                     3   j = 13                   5 1   j = 11                 8 3     j = 9               10 5       j = 7             10 8         j = 5           9 10           j = 3         8 10             j = 1       5 10               j = -1     2 7                 j = -3   1 5                   j = -5   2                     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, 78]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 78]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[12, 6, 13, 5], X[14, 7, 15, 8], > X[20, 10, 21, 9], X[2, 11, 3, 12], X[16, 14, 17, 13], X[6, 15, 7, 16], > X[22, 18, 1, 17], X[8, 20, 9, 19], X[18, 22, 19, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 78]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -8, 4, -10, 5, -2, 6, -3, 7, -4, 8, -7, 9, -11, 10, > -5, 11, -9]
In[5]:=	DTCode[Knot[11, Alternating, 78]]
Out[5]=	DTCode[4, 10, 12, 14, 20, 2, 16, 6, 22, 8, 18]
In[6]:=	alex = Alexander[Knot[11, Alternating, 78]][t]
Out[6]=	2 12 29 2 3 -37 + -- - -- + -- + 29 t - 12 t + 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 78]][z]
Out[7]=	2 6 1 - z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 78], Knot[11, Alternating, 130]}
In[9]:=	{KnotDet[Knot[11, Alternating, 78]], KnotSignature[Knot[11, Alternating, 78]]}
Out[9]=	{123, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 78]][q]
Out[10]=	-3 3 7 2 3 4 5 6 7 8 -12 + q - -- + - + 17 q - 19 q + 20 q - 18 q + 13 q - 8 q + 4 q - q 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 78]}
In[12]:=	A2Invariant[Knot[11, Alternating, 78]][q]
Out[12]=	-10 -6 3 2 2 4 6 8 12 14 16 -1 + q - q + -- - -- + 3 q - 3 q + 4 q - q + q - 4 q + 4 q - 4 2 q q 18 20 22 24 > q - q + 2 q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 78]][a, z]
Out[13]=	2 2 4 4 4 6 6 -2 2 2 z 2 z 2 2 4 z 2 z z z z -1 + a + a - 3 z - -- + ---- + a z - 2 z - -- + ---- + -- + -- + -- 6 4 6 4 2 4 2 a a a a a a a
In[14]:=	Kauffman[Knot[11, Alternating, 78]][a, z]
Out[14]=	2 2 2 -2 2 z 4 z 6 z 5 z 2 2 z 3 z z -1 - a - a - -- - --- - --- - --- - 2 a z + 5 z + ---- + ---- - -- + 7 5 3 a 8 6 4 a a a a a a 3 3 3 3 3 4 4 2 2 z 6 z 19 z 22 z 17 z 3 4 6 z 3 z > 3 a z - -- + ---- + ----- + ----- + ----- + 7 a z - 2 z - ---- - ---- + 9 7 5 3 a 8 6 a a a a a a 4 4 5 5 5 5 5 11 z 9 z 2 4 z 11 z 24 z 23 z 19 z 5 > ----- + ---- - 3 a z + -- - ----- - ----- - ----- - ----- - 8 a z - 4 2 9 7 5 3 a a a a a a a 6 6 6 6 7 7 7 6 4 z 6 z 24 z 21 z 2 6 7 z 7 z 3 z 7 > 6 z + ---- - ---- - ----- - ----- + a z + ---- + ---- + ---- + 3 a z + 8 6 4 2 7 5 a a a a a a a 8 8 8 9 9 9 10 10 8 7 z 13 z 10 z 4 z 7 z 3 z z z > 4 z + ---- + ----- + ----- + ---- + ---- + ---- + --- + --- 6 4 2 5 3 a 4 2 a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 78]], Vassiliev[3][Knot[11, Alternating, 78]]}
Out[15]=	{-1, 0}
In[16]:=	Kh[Knot[11, Alternating, 78]][q, t]
Out[16]=	3 1 2 1 5 2 7 5 q 3 10 q + 8 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 10 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 > 9 q t + 10 q t + 10 q t + 8 q t + 10 q t + 5 q t + 8 q t + 11 5 13 5 13 6 15 6 17 7 > 3 q t + 5 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a78

K11a77 K11a79