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DrawMorseLink

PD Presentation:

X10,1,11,2 X18,11,19,12 X8,9,1,10 X22,19,9,20 X20,6,21,5 X4,22,5,21 X14,7,15,8 X12,4,13,3 X16,14,17,13 X6,15,7,16 X2,18,3,17

Gauss Code:

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

Jones Polynomial:

- q-13/2 + 4q-11/2 - 9q-9/2 + 16q-7/2 - 22q-5/2 + 24q-3/2 - 25q-1/2 + 21q1/2 - 16q3/2 + 9q5/2 - 4q7/2 + q9/2

A2 (sl(3)) Invariant:

q-20 - q-18 - q-16 + 3q-14 - 5q-12 + 2q-10 + 2q-8 - 2q-6 + 6q-4 - 3q-2 + 5 - q4 + 5q6 - 3q8 + q10 + q12 - q14

HOMFLY-PT Polynomial:

a-3z + a-3z3 - a-1z-1 - 3a-1z - 4a-1z3 - 2a-1z5 + az-1 + 4az + 5az3 + 3az5 + az7 - 3a3z - 4a3z3 - 2a3z5 + a5z + a5z3

Kauffman Polynomial:

- a-4z2 + 2a-4z4 - a-4z6 + 2a-3z - 7a-3z3 + 9a-3z5 - 4a-3z7 + a-2z2 - 8a-2z4 + 14a-2z6 - 7a-2z8 - a-1z-1 + 6a-1z - 18a-1z3 + 18a-1z5 + 2a-1z7 - 6a-1z9 + 1 + 6z2 - 28z4 + 42z6 - 16z8 - 2z10 - az-1 + 8az - 23az3 + 20az5 + 11az7 - 13az9 + 5a2z2 - 27a2z4 + 43a2z6 - 19a2z8 - 2a2z10 + 6a3z - 20a3z3 + 23a3z5 - 3a3z7 - 7a3z9 - a4z2 - 4a4z4 + 12a4z6 - 10a4z8 + 2a5z - 7a5z3 + 11a5z5 - 8a5z7 - 2a6z2 + 5a6z4 - 4a6z6 + a7z3 - a7z5

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 10                       1 j = 8                     3   j = 6                   6 1   j = 4                 10 3     j = 2               11 6       j = 0             14 10         j = -2           12 13           j = -4         10 12             j = -6       6 12               j = -8     3 10                 j = -10   1 6                   j = -12   3                     j = -14 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]:=	Length[Skeleton[L]]
Out[2]=	2
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 323]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 323]]
Out[4]=	PD[X[10, 1, 11, 2], X[18, 11, 19, 12], X[8, 9, 1, 10], X[22, 19, 9, 20], > X[20, 6, 21, 5], X[4, 22, 5, 21], X[14, 7, 15, 8], X[12, 4, 13, 3], > X[16, 14, 17, 13], X[6, 15, 7, 16], X[2, 18, 3, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 8, -6, 5, -10, 7, -3}, > {3, -1, 2, -8, 9, -7, 10, -9, 11, -2, 4, -5, 6, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(13/2) 4 9 16 22 24 25 -q + ----- - ---- + ---- - ---- + ---- - ------- + 21 Sqrt[q] - 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q 3/2 5/2 7/2 9/2 > 16 q + 9 q - 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 -18 -16 3 5 2 2 2 6 3 4 6 5 + q - q - q + --- - --- + --- + -- - -- + -- - -- - q + 5 q - 14 12 10 8 6 4 2 q q q q q q q 8 10 12 14 > 3 q + q + q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 323]][a, z]
Out[8]=	3 3 1 a z 3 z 3 5 z 4 z 3 3 3 -(---) + - + -- - --- + 4 a z - 3 a z + a z + -- - ---- + 5 a z - 4 a z + a z z 3 a 3 a a a 5 5 3 2 z 5 3 5 7 > a z - ---- + 3 a z - 2 a z + a z a
In[9]:=	Kauffman[Link[11, Alternating, 323]][a, z]
Out[9]=	2 2 1 a 2 z 6 z 3 5 2 z z 2 2 1 - --- - - + --- + --- + 8 a z + 6 a z + 2 a z + 6 z - -- + -- + 5 a z - a z z 3 a 4 2 a a a 3 3 4 2 6 2 7 z 18 z 3 3 3 5 3 7 3 > a z - 2 a z - ---- - ----- - 23 a z - 20 a z - 7 a z + a z - 3 a a 4 4 5 5 4 2 z 8 z 2 4 4 4 6 4 9 z 18 z > 28 z + ---- - ---- - 27 a z - 4 a z + 5 a z + ---- + ----- + 4 2 3 a a a a 6 6 5 3 5 5 5 7 5 6 z 14 z 2 6 > 20 a z + 23 a z + 11 a z - a z + 42 z - -- + ----- + 43 a z + 4 2 a a 7 7 4 6 6 6 4 z 2 z 7 3 7 5 7 8 > 12 a z - 4 a z - ---- + ---- + 11 a z - 3 a z - 8 a z - 16 z - 3 a a 8 9 7 z 2 8 4 8 6 z 9 3 9 10 2 10 > ---- - 19 a z - 10 a z - ---- - 13 a z - 7 a z - 2 z - 2 a z 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	13 1 3 1 6 3 10 6 12 14 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 2 14 6 12 5 10 5 10 4 8 4 8 3 6 3 6 2 q q t q t q t q t q t q t q t q t 10 12 12 2 2 2 4 2 4 3 > ----- + ---- + ---- + 10 t + 11 q t + 6 q t + 10 q t + 3 q t + 4 2 4 2 q t q t q t 6 3 6 4 8 4 10 5 > 6 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a323

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