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

K11a82 K11a84

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 12 16 2 22 18 20 8 14 6

Alexander Polynomial:

t-4 - 5t-3 + 14t-2 - 23t-1 + 27 - 23t + 14t2 - 5t3 + t4

Conway Polynomial:

1 + 4z2 + 4z4 + 3z6 + z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{113, 4}

Jones Polynomial:

- q-1 + 3 - 6q + 11q2 - 14q3 + 18q4 - 18q5 + 16q6 - 13q7 + 8q8 - 4q9 + q10

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

{...}

A2 (sl(3)) Invariant:

- q-2 + 1 - 2q2 + q4 + 2q6 - q8 + 6q10 - q12 + 3q14 - 3q18 + q20 - 4q22 + q24 - q28 + q30

HOMFLY-PT Polynomial:

a-8 + 2a-8z2 + a-8z4 - 6a-6 - 11a-6z2 - 8a-6z4 - 2a-6z6 + 8a-4 + 18a-4z2 + 15a-4z4 + 6a-4z6 + a-4z8 - 2a-2 - 5a-2z2 - 4a-2z4 - a-2z6

Kauffman Polynomial:

a-12z4 - 2a-11z3 + 4a-11z5 + a-10z2 - 7a-10z4 + 8a-10z6 - a-9z + 7a-9z3 - 15a-9z5 + 11a-9z7 + a-8 - 3a-8z2 + 6a-8z4 - 14a-8z6 + 10a-8z8 - 9a-7z + 29a-7z3 - 32a-7z5 + 5a-7z7 + 5a-7z9 + 6a-6 - 24a-6z2 + 50a-6z4 - 51a-6z6 + 16a-6z8 + a-6z10 - 13a-5z + 31a-5z3 - 15a-5z5 - 13a-5z7 + 8a-5z9 + 8a-4 - 29a-4z2 + 52a-4z4 - 41a-4z6 + 9a-4z8 + a-4z10 - 7a-3z + 16a-3z3 - 6a-3z5 - 6a-3z7 + 3a-3z9 + 2a-2 - 9a-2z2 + 16a-2z4 - 12a-2z6 + 3a-2z8 - 2a-1z + 5a-1z3 - 4a-1z5 + a-1z7

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

{4, 6}

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=4 is the signature of 1183. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 j = 21                       1 j = 19                     3   j = 17                   5 1   j = 15                 8 3     j = 13               8 5       j = 11             10 8         j = 9           8 8           j = 7         6 10             j = 5       5 8               j = 3     2 7                 j = 1   1 4                   j = -1   2                     j = -3 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, 83]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 83]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[12, 5, 13, 6], X[16, 8, 17, 7], > X[2, 10, 3, 9], X[22, 11, 1, 12], X[18, 14, 19, 13], X[20, 16, 21, 15], > X[8, 18, 9, 17], X[14, 20, 15, 19], X[6, 21, 7, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 83]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -11, 4, -9, 5, -2, 6, -3, 7, -10, 8, -4, 9, -7, 10, > -8, 11, -6]
In[5]:=	DTCode[Knot[11, Alternating, 83]]
Out[5]=	DTCode[4, 10, 12, 16, 2, 22, 18, 20, 8, 14, 6]
In[6]:=	alex = Alexander[Knot[11, Alternating, 83]][t]
Out[6]=	-4 5 14 23 2 3 4 27 + t - -- + -- - -- - 23 t + 14 t - 5 t + t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 83]][z]
Out[7]=	2 4 6 8 1 + 4 z + 4 z + 3 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 83]}
In[9]:=	{KnotDet[Knot[11, Alternating, 83]], KnotSignature[Knot[11, Alternating, 83]]}
Out[9]=	{113, 4}
In[10]:=	J=Jones[Knot[11, Alternating, 83]][q]
Out[10]=	1 2 3 4 5 6 7 8 9 10 3 - - - 6 q + 11 q - 14 q + 18 q - 18 q + 16 q - 13 q + 8 q - 4 q + q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 83]}
In[12]:=	A2Invariant[Knot[11, Alternating, 83]][q]
Out[12]=	-2 2 4 6 8 10 12 14 18 20 22 1 - q - 2 q + q + 2 q - q + 6 q - q + 3 q - 3 q + q - 4 q + 24 28 30 > q - q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 83]][a, z]
Out[13]=	2 2 2 2 4 4 4 4 -8 6 8 2 2 z 11 z 18 z 5 z z 8 z 15 z 4 z a - -- + -- - -- + ---- - ----- + ----- - ---- + -- - ---- + ----- - ---- - 6 4 2 8 6 4 2 8 6 4 2 a a a a a a a a a a a 6 6 6 8 2 z 6 z z z > ---- + ---- - -- + -- 6 4 2 4 a a a a
In[14]:=	Kauffman[Knot[11, Alternating, 83]][a, z]
Out[14]=	2 2 2 2 -8 6 8 2 z 9 z 13 z 7 z 2 z z 3 z 24 z 29 z a + -- + -- + -- - -- - --- - ---- - --- - --- + --- - ---- - ----- - ----- - 6 4 2 9 7 5 3 a 10 8 6 4 a a a a a a a a a a a 2 3 3 3 3 3 3 4 4 4 9 z 2 z 7 z 29 z 31 z 16 z 5 z z 7 z 6 z > ---- - ---- + ---- + ----- + ----- + ----- + ---- + --- - ---- + ---- + 2 11 9 7 5 3 a 12 10 8 a a a a a a a a a 4 4 4 5 5 5 5 5 5 6 50 z 52 z 16 z 4 z 15 z 32 z 15 z 6 z 4 z 8 z > ----- + ----- + ----- + ---- - ----- - ----- - ----- - ---- - ---- + ---- - 6 4 2 11 9 7 5 3 a 10 a a a a a a a a a 6 6 6 6 7 7 7 7 7 8 14 z 51 z 41 z 12 z 11 z 5 z 13 z 6 z z 10 z > ----- - ----- - ----- - ----- + ----- + ---- - ----- - ---- + -- + ----- + 8 6 4 2 9 7 5 3 a 8 a a a a a a a a a 8 8 8 9 9 9 10 10 16 z 9 z 3 z 5 z 8 z 3 z z z > ----- + ---- + ---- + ---- + ---- + ---- + --- + --- 6 4 2 7 5 3 6 4 a a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 83]], Vassiliev[3][Knot[11, Alternating, 83]]}
Out[15]=	{4, 6}
In[16]:=	Kh[Knot[11, Alternating, 83]][q, t]
Out[16]=	3 3 5 1 2 q 4 q 2 q 5 7 7 2 7 q + 5 q + ----- + ---- + -- + --- + ---- + 8 q t + 6 q t + 10 q t + 3 3 2 2 t t q t q t t 9 2 9 3 11 3 11 4 13 4 13 5 15 5 > 8 q t + 8 q t + 10 q t + 8 q t + 8 q t + 5 q t + 8 q t + 15 6 17 6 17 7 19 7 21 8 > 3 q t + 5 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a83

K11a82 K11a84