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DrawMorseLink

PD Presentation:

X8192 X16,5,17,6 X18,9,19,10 X10,17,11,18 X20,15,21,16 X14,4,15,3 X4,22,5,21 X2738 X22,12,13,11 X12,14,7,13 X6,19,1,20

Gauss Code:

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

Jones Polynomial:

- q-7 + 4q-6 - 9q-5 + 15q-4 - 19q-3 + 24q-2 - 22q-1 + 21 - 15q + 9q2 - 4q3 + q4

A2 (sl(3)) Invariant:

- q-22 + 2q-18 - 3q-16 + 3q-14 + 2q-12 + 9q-8 + 2q-6 + 7q-4 + 3q-2 + 5q2 - 4q4 + 2q6 + q8 - 2q10 + q12

HOMFLY-PT Polynomial:

a-2z2 + a-2z4 + z-2 + 2 + z2 - z4 - z6 - 2a2z-2 - 4a2 - 7a2z2 - 6a2z4 - 2a2z6 + a4z-2 + 3a4 + 6a4z2 + 3a4z4 - a6 - a6z2

Kauffman Polynomial:

a-4z4 - a-3z3 + 4a-3z5 + 2a-2z2 - 7a-2z4 + 9a-2z6 + 7a-1z3 - 18a-1z5 + 14a-1z7 - z-2 + 3 - 9z2 + 20z4 - 29z6 + 16z8 + 2az-1 - 2az + 8az3 - 8az5 - 13az7 + 11az9 - 2a2z-2 + 7a2 - 33a2z2 + 74a2z4 - 73a2z6 + 19a2z8 + 3a2z10 + 2a3z-1 - 4a3z + 3a3z3 + 24a3z5 - 44a3z7 + 17a3z9 - a4z-2 + 7a4 - 29a4z2 + 60a4z4 - 48a4z6 + 7a4z8 + 3a4z10 - 3a5z + 6a5z3 + 7a5z5 - 16a5z7 + 6a5z9 + 2a6 - 7a6z2 + 14a6z4 - 13a6z6 + 4a6z8 - a7z + 3a7z3 - 3a7z5 + a7z7

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 9                       1 j = 7                     3   j = 5                   6 1   j = 3                 9 3     j = 1               12 6       j = -1             12 11         j = -3           12 10           j = -5         9 14             j = -7       6 10               j = -9     3 9                 j = -11   1 6                   j = -13   3                     j = -15 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]=	3
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 516]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 516]]
Out[4]=	PD[X[8, 1, 9, 2], X[16, 5, 17, 6], X[18, 9, 19, 10], X[10, 17, 11, 18], > X[20, 15, 21, 16], X[14, 4, 15, 3], X[4, 22, 5, 21], X[2, 7, 3, 8], > X[22, 12, 13, 11], X[12, 14, 7, 13], X[6, 19, 1, 20]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -8, 6, -7, 2, -11}, {8, -1, 3, -4, 9, -10}, > {10, -6, 5, -2, 4, -3, 11, -5, 7, -9}]
In[6]:=	Jones[L][q]
Out[6]=	-7 4 9 15 19 24 22 2 3 4 21 - q + -- - -- + -- - -- + -- - -- - 15 q + 9 q - 4 q + q 6 5 4 3 2 q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 2 3 3 2 9 2 7 3 2 4 6 8 -q + --- - --- + --- + --- + -- + -- + -- + -- + 5 q - 4 q + 2 q + q - 18 16 14 12 8 6 4 2 q q q q q q q q 10 12 > 2 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 516]][a, z]
Out[8]=	2 4 2 2 4 6 -2 2 a a 2 z 2 2 4 2 6 2 2 - 4 a + 3 a - a + z - ---- + -- + z + -- - 7 a z + 6 a z - a z - 2 2 2 z z a 4 4 z 2 4 4 4 6 2 6 > z + -- - 6 a z + 3 a z - z - 2 a z 2 a
In[9]:=	Kauffman[Link[11, Alternating, 516]][a, z]
Out[9]=	2 4 3 2 4 6 -2 2 a a 2 a 2 a 3 3 + 7 a + 7 a + 2 a - z - ---- - -- + --- + ---- - 2 a z - 4 a z - 2 2 z z z z 2 3 3 5 7 2 2 z 2 2 4 2 6 2 z 7 z > 3 a z - a z - 9 z + ---- - 33 a z - 29 a z - 7 a z - -- + ---- + 2 3 a a a 4 4 3 3 3 5 3 7 3 4 z 7 z 2 4 > 8 a z + 3 a z + 6 a z + 3 a z + 20 z + -- - ---- + 74 a z + 4 2 a a 5 5 4 4 6 4 4 z 18 z 5 3 5 5 5 > 60 a z + 14 a z + ---- - ----- - 8 a z + 24 a z + 7 a z - 3 a a 6 7 7 5 6 9 z 2 6 4 6 6 6 14 z 7 > 3 a z - 29 z + ---- - 73 a z - 48 a z - 13 a z + ----- - 13 a z - 2 a a 3 7 5 7 7 7 8 2 8 4 8 6 8 > 44 a z - 16 a z + a z + 16 z + 19 a z + 7 a z + 4 a z + 9 3 9 5 9 2 10 4 10 > 11 a z + 17 a z + 6 a z + 3 a z + 3 a z
In[10]:=	Kh[L][q, t]
Out[10]=	11 1 3 1 6 3 9 6 10 -- + 12 q + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + q 15 7 13 6 11 6 11 5 9 5 9 4 7 4 7 3 q t q t q t q t q t q t q t q t 9 14 12 10 12 3 3 2 5 2 > ----- + ----- + ----- + ---- + --- + 6 q t + 9 q t + 3 q t + 6 q t + 5 3 5 2 3 2 3 q t q t q t q t q t 5 3 7 3 9 4 > q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a516

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