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DrawMorseLink

PD Presentation:

X6172 X10,4,11,3 X18,7,19,8 X14,21,15,22 X20,10,21,9 X8,13,9,14 X22,15,17,16 X16,17,5,18 X12,20,13,19 X2536 X4,12,1,11

Gauss Code:

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

Jones Polynomial:

- q-8 + 3q-7 - 7q-6 + 13q-5 - 17q-4 + 21q-3 - 20q-2 + 19q-1 - 13 + 9q - 4q2 + q3

A2 (sl(3)) Invariant:

- q-24 - 2q-18 + 4q-16 - q-14 + 4q-12 + 5q-10 + q-8 + 8q-6 - q-4 + 6q-2 + 2 + 3q4 - 2q6 + q8

HOMFLY-PT Polynomial:

z-2 + 3 + 3z2 + 3z4 + z6 - 2a2z-2 - 7a2 - 11a2z2 - 10a2z4 - 5a2z6 - a2z8 + a4z-2 + 6a4 + 11a4z2 + 8a4z4 + 2a4z6 - 2a6 - 3a6z2 - a6z4

Kauffman Polynomial:

- 2a-2z4 + a-2z6 + 3a-1z3 - 9a-1z5 + 4a-1z7 - z-2 + 4 - 10z2 + 21z4 - 23z6 + 8z8 + 2az-1 - 3az + az3 + 7az5 - 15az7 + 7az9 - 2a2z-2 + 9a2 - 30a2z2 + 56a2z4 - 47a2z6 + 12a2z8 + 2a2z10 + 2a3z-1 - 6a3z + a3z3 + 18a3z5 - 26a3z7 + 12a3z9 - a4z-2 + 7a4 - 23a4z2 + 39a4z4 - 32a4z6 + 10a4z8 + 2a4z10 - 3a5z + 6a5z3 - 5a5z5 - 2a5z7 + 5a5z9 + a6z4 - 6a6z6 + 6a6z8 + a7z + a7z3 - 6a7z5 + 5a7z7 - a8 + 3a8z2 - 5a8z4 + 3a8z6 + a9z - 2a9z3 + a9z5

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 = 7                       1 j = 5                     3   j = 3                   6 1   j = 1                 7 3     j = -1               12 6       j = -3             10 9         j = -5           11 10           j = -7         8 12             j = -9       5 9               j = -11     2 8                 j = -13   1 5                   j = -15   2                     j = -17 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, 480]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 480]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 4, 11, 3], X[18, 7, 19, 8], X[14, 21, 15, 22], > X[20, 10, 21, 9], X[8, 13, 9, 14], X[22, 15, 17, 16], X[16, 17, 5, 18], > X[12, 20, 13, 19], X[2, 5, 3, 6], X[4, 12, 1, 11]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {8, -3, 9, -5, 4, -7}, > {10, -1, 3, -6, 5, -2, 11, -9, 6, -4, 7, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-8 3 7 13 17 21 20 19 2 3 -13 - q + -- - -- + -- - -- + -- - -- + -- + 9 q - 4 q + q 7 6 5 4 3 2 q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 2 4 -14 4 5 -8 8 -4 6 4 6 8 2 - q - --- + --- - q + --- + --- + q + -- - q + -- + 3 q - 2 q + q 18 16 12 10 6 2 q q q q q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 480]][a, z]
Out[8]=	2 4 2 4 6 -2 2 a a 2 2 2 4 2 3 - 7 a + 6 a - 2 a + z - ---- + -- + 3 z - 11 a z + 11 a z - 2 2 z z 6 2 4 2 4 4 4 6 4 6 2 6 4 6 2 8 > 3 a z + 3 z - 10 a z + 8 a z - a z + z - 5 a z + 2 a z - a z
In[9]:=	Kauffman[Link[11, Alternating, 480]][a, z]
Out[9]=	2 4 3 2 4 8 -2 2 a a 2 a 2 a 3 5 4 + 9 a + 7 a - a - z - ---- - -- + --- + ---- - 3 a z - 6 a z - 3 a z + 2 2 z z z z 3 7 9 2 2 2 4 2 8 2 3 z 3 3 3 > a z + a z - 10 z - 30 a z - 23 a z + 3 a z + ---- + a z + a z + a 4 5 3 7 3 9 3 4 2 z 2 4 4 4 6 4 > 6 a z + a z - 2 a z + 21 z - ---- + 56 a z + 39 a z + a z - 2 a 5 8 4 9 z 5 3 5 5 5 7 5 9 5 6 > 5 a z - ---- + 7 a z + 18 a z - 5 a z - 6 a z + a z - 23 z + a 6 7 z 2 6 4 6 6 6 8 6 4 z 7 3 7 > -- - 47 a z - 32 a z - 6 a z + 3 a z + ---- - 15 a z - 26 a z - 2 a a 5 7 7 7 8 2 8 4 8 6 8 9 > 2 a z + 5 a z + 8 z + 12 a z + 10 a z + 6 a z + 7 a z + 3 9 5 9 2 10 4 10 > 12 a z + 5 a z + 2 a z + 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	9 12 1 2 1 5 2 8 5 9 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 3 q 17 7 15 6 13 6 13 5 11 5 11 4 9 4 9 3 q q t q t q t q t q t q t q t q t 8 12 11 10 10 6 t 2 3 2 > ----- + ----- + ----- + ---- + ---- + --- + 7 q t + 3 q t + 6 q t + 7 3 7 2 5 2 5 3 q q t q t q t q t q t 3 3 5 3 7 4 > q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a480

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