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

K11a85 K11a87

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 12 16 18 2 20 6 8 22 14

Alexander Polynomial:

- t-4 + 5t-3 - 12t-2 + 18t-1 - 19 + 18t - 12t2 + 5t3 - t4

Conway Polynomial:

1 - z2 - 2z4 - 3z6 - z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{91, 2}

Jones Polynomial:

- q-4 + 3q-3 - 5q-2 + 9q-1 - 12 + 14q - 14q2 + 13q3 - 10q4 + 6q5 - 3q6 + q7

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

{K11a205, ...}

A2 (sl(3)) Invariant:

- q-12 + q-8 + 3q-4 - q-2 + 1 + q2 - 2q4 + 3q6 - 3q8 + q10 - q12 - q14 + 2q16 - q18 + q20

HOMFLY-PT Polynomial:

2a-4 + 5a-4z2 + 4a-4z4 + a-4z6 - 5a-2 - 15a-2z2 - 14a-2z4 - 6a-2z6 - a-2z8 + 5 + 12z2 + 9z4 + 2z6 - a2 - 3a2z2 - a2z4

Kauffman Polynomial:

- a-8z2 + a-8z4 + a-7z - 3a-7z3 + 3a-7z5 + 2a-6z2 - 5a-6z4 + 5a-6z6 + a-5z + a-5z3 - 6a-5z5 + 6a-5z7 + 2a-4 - 5a-4z2 + 7a-4z4 - 9a-4z6 + 6a-4z8 + a-3z3 - 2a-3z5 - 4a-3z7 + 4a-3z9 + 5a-2 - 26a-2z2 + 45a-2z4 - 36a-2z6 + 9a-2z8 + a-2z10 - a-1z - 2a-1z3 + 14a-1z5 - 20a-1z7 + 7a-1z9 + 5 - 26z2 + 49z4 - 35z6 + 6z8 + z10 - 2az + 5az3 + 3az5 - 9az7 + 3az9 + a2 - 8a2z2 + 17a2z4 - 13a2z6 + 3a2z8 - a3z + 4a3z3 - 4a3z5 + a3z7

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

{-1, -2}

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 1186. 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 = 15                       1 j = 13                     2   j = 11                   4 1   j = 9                 6 2     j = 7               7 4       j = 5             7 6         j = 3           7 7           j = 1         6 8             j = -1       3 6               j = -3     2 6                 j = -5   1 3                   j = -7   2                     j = -9 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, 86]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 86]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[12, 6, 13, 5], X[16, 8, 17, 7], > X[18, 10, 19, 9], X[2, 11, 3, 12], X[20, 13, 21, 14], X[6, 16, 7, 15], > X[8, 18, 9, 17], X[22, 19, 1, 20], X[14, 21, 15, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 86]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -8, 4, -9, 5, -2, 6, -3, 7, -11, 8, -4, 9, -5, 10, > -7, 11, -10]
In[5]:=	DTCode[Knot[11, Alternating, 86]]
Out[5]=	DTCode[4, 10, 12, 16, 18, 2, 20, 6, 8, 22, 14]
In[6]:=	alex = Alexander[Knot[11, Alternating, 86]][t]
Out[6]=	-4 5 12 18 2 3 4 -19 - t + -- - -- + -- + 18 t - 12 t + 5 t - t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 86]][z]
Out[7]=	2 4 6 8 1 - z - 2 z - 3 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 86]}
In[9]:=	{KnotDet[Knot[11, Alternating, 86]], KnotSignature[Knot[11, Alternating, 86]]}
Out[9]=	{91, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 86]][q]
Out[10]=	-4 3 5 9 2 3 4 5 6 7 -12 - q + -- - -- + - + 14 q - 14 q + 13 q - 10 q + 6 q - 3 q + q 3 2 q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 86], Knot[11, Alternating, 205]}
In[12]:=	A2Invariant[Knot[11, Alternating, 86]][q]
Out[12]=	-12 -8 3 -2 2 4 6 8 10 12 14 16 1 - q + q + -- - q + q - 2 q + 3 q - 3 q + q - q - q + 2 q - 4 q 18 20 > q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 86]][a, z]
Out[13]=	2 2 4 4 2 5 2 2 5 z 15 z 2 2 4 4 z 14 z 5 + -- - -- - a + 12 z + ---- - ----- - 3 a z + 9 z + ---- - ----- - 4 2 4 2 4 2 a a a a a a 6 6 8 2 4 6 z 6 z z > a z + 2 z + -- - ---- - -- 4 2 2 a a a
In[14]:=	Kauffman[Knot[11, Alternating, 86]][a, z]
Out[14]=	2 2 2 2 5 2 z z z 3 2 z 2 z 5 z 5 + -- + -- + a + -- + -- - - - 2 a z - a z - 26 z - -- + ---- - ---- - 4 2 7 5 a 8 6 4 a a a a a a a 2 3 3 3 3 4 26 z 2 2 3 z z z 2 z 3 3 3 4 z > ----- - 8 a z - ---- + -- + -- - ---- + 5 a z + 4 a z + 49 z + -- - 2 7 5 3 a 8 a a a a a 4 4 4 5 5 5 5 5 z 7 z 45 z 2 4 3 z 6 z 2 z 14 z 5 > ---- + ---- + ----- + 17 a z + ---- - ---- - ---- + ----- + 3 a z - 6 4 2 7 5 3 a a a a a a a 6 6 6 7 7 7 3 5 6 5 z 9 z 36 z 2 6 6 z 4 z 20 z > 4 a z - 35 z + ---- - ---- - ----- - 13 a z + ---- - ---- - ----- - 6 4 2 5 3 a a a a a a 8 8 9 9 7 3 7 8 6 z 9 z 2 8 4 z 7 z 9 > 9 a z + a z + 6 z + ---- + ---- + 3 a z + ---- + ---- + 3 a z + 4 2 3 a a a a 10 10 z > z + --- 2 a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 86]], Vassiliev[3][Knot[11, Alternating, 86]]}
Out[15]=	{-1, -2}
In[16]:=	Kh[Knot[11, Alternating, 86]][q, t]
Out[16]=	3 1 2 1 3 2 6 3 6 6 q 8 q + 7 q + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + --- + 9 5 7 4 5 4 5 3 3 3 3 2 2 q t t q t q t q t q t q t q t q t 3 5 5 2 7 2 7 3 9 3 9 4 > 7 q t + 7 q t + 6 q t + 7 q t + 4 q t + 6 q t + 2 q t + 11 4 11 5 13 5 15 6 > 4 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a86

K11a85 K11a87