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DrawMorseLink

PD Presentation:

X8192 X2,9,3,10 X10,3,11,4 X14,5,15,6 X20,18,21,17 X18,12,19,11 X12,20,13,19 X22,16,7,15 X16,22,17,21 X6718 X4,13,5,14

Gauss Code:

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

Jones Polynomial:

- q-15/2 + 2q-13/2 - 4q-11/2 + 6q-9/2 - 8q-7/2 + 8q-5/2 - 9q-3/2 + 7q-1/2 - 6q1/2 + 4q3/2 - 2q5/2 + q7/2

A2 (sl(3)) Invariant:

q-22 + q-18 + q-16 + 2q-12 + 3q-8 + 2q-6 + q-4 + 2q-2 - 1 - q4 - q6 - q10

HOMFLY-PT Polynomial:

a-1z-1 + 4a-1z + 4a-1z3 + a-1z5 - 2az-1 - 6az - 8az3 - 5az5 - az7 - 5a3z - 8a3z3 - 5a3z5 - a3z7 + a5z-1 + 4a5z + 4a5z3 + a5z5

Kauffman Polynomial:

- 2a-2 + 9a-2z2 - 12a-2z4 + 6a-2z6 - a-2z8 + a-1z-1 - 4a-1z + 12a-1z3 - 19a-1z5 + 11a-1z7 - 2a-1z9 - 5 + 23z2 - 33z4 + 14z6 + z8 - z10 + 2az-1 - 8az + 19az3 - 32az5 + 23az7 - 5az9 - 3a2 + 12a2z2 - 26a2z4 + 20a2z6 - 2a2z8 - a2z10 + 4a3z - 7a3z3 + 2a3z5 + 7a3z7 - 3a3z9 + a4 - 5a4z2 + 2a4z4 + 8a4z6 - 4a4z8 - a5z-1 + 6a5z - 11a5z3 + 12a5z5 - 5a5z7 - 2a6z2 + 5a6z4 - 4a6z6 - a7z + 2a7z3 - 3a7z5 + a8z2 - 2a8z4 + a9z - a9z3

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 = 8                       1 j = 6                     1   j = 4                   3 1   j = 2                 3 1     j = 0               4 3       j = -2             5 3         j = -4           4 5           j = -6         4 4             j = -8       2 4               j = -10     2 4                 j = -12     2                   j = -14 1 2                     j = -16 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, 165]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 165]]
Out[4]=	PD[X[8, 1, 9, 2], X[2, 9, 3, 10], X[10, 3, 11, 4], X[14, 5, 15, 6], > X[20, 18, 21, 17], X[18, 12, 19, 11], X[12, 20, 13, 19], X[22, 16, 7, 15], > X[16, 22, 17, 21], X[6, 7, 1, 8], X[4, 13, 5, 14]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -11, 4, -10}, > {10, -1, 2, -3, 6, -7, 11, -4, 8, -9, 5, -6, 7, -5, 9, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-(15/2) 2 4 6 8 8 9 7 -q + ----- - ----- + ---- - ---- + ---- - ---- + ------- - 6 Sqrt[q] + 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q 3/2 5/2 7/2 > 4 q - 2 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 -18 -16 2 3 2 -4 2 4 6 10 -1 + q + q + q + --- + -- + -- + q + -- - q - q - q 12 8 6 2 q q q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 165]][a, z]
Out[8]=	5 3 1 2 a a 4 z 3 5 4 z 3 3 3 --- - --- + -- + --- - 6 a z - 5 a z + 4 a z + ---- - 8 a z - 8 a z + a z z z a a 5 5 3 z 5 3 5 5 5 7 3 7 > 4 a z + -- - 5 a z - 5 a z + a z - a z - a z a
In[9]:=	Kauffman[Link[11, Alternating, 165]][a, z]
Out[9]=	5 2 2 4 1 2 a a 4 z 3 5 7 -5 - -- - 3 a + a + --- + --- - -- - --- - 8 a z + 4 a z + 6 a z - a z + 2 a z z z a a 2 3 9 2 9 z 2 2 4 2 6 2 8 2 12 z > a z + 23 z + ---- + 12 a z - 5 a z - 2 a z + a z + ----- + 2 a a 4 3 3 3 5 3 7 3 9 3 4 12 z 2 4 > 19 a z - 7 a z - 11 a z + 2 a z - a z - 33 z - ----- - 26 a z + 2 a 5 4 4 6 4 8 4 19 z 5 3 5 5 5 > 2 a z + 5 a z - 2 a z - ----- - 32 a z + 2 a z + 12 a z - a 6 7 7 5 6 6 z 2 6 4 6 6 6 11 z 7 > 3 a z + 14 z + ---- + 20 a z + 8 a z - 4 a z + ----- + 23 a z + 2 a a 8 9 3 7 5 7 8 z 2 8 4 8 2 z 9 3 9 > 7 a z - 5 a z + z - -- - 2 a z - 4 a z - ---- - 5 a z - 3 a z - 2 a a 10 2 10 > z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	5 5 1 1 2 2 2 4 2 4 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 4 2 16 6 14 6 14 5 12 4 10 4 10 3 8 3 8 2 q q q t q t q t q t q t q t q t q t 4 4 4 3 t 2 2 2 2 3 4 3 > ----- + ---- + ---- + 4 t + --- + 3 t + 3 q t + q t + 3 q t + 6 2 6 4 2 q t q t q t q 4 4 6 4 8 5 > q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a165

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