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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-7 + 3q-6 - 7q-5 + 13q-4 - 16q-3 + 21q-2 - 20q-1 + 19 - 14q + 9q2 - 4q3 + q4

A2 (sl(3)) Invariant:

- q-22 + q-18 - 3q-16 + 3q-14 + 3q-12 + q-10 + 9q-8 + 3q-6 + 6q-4 + 3q-2 - 1 + 4q2 - 4q4 + 2q6 + 2q8 - 2q10 + q12

HOMFLY-PT Polynomial:

a-2 + a-2z2 + a-2z4 + z-2 - 1 - 3z2 - 2z4 - z6 - 2a2z-2 - a2 - a2z4 - a2z6 + a4z-2 + 2a4 + 3a4z2 + 2a4z4 - a6 - a6z2

Kauffman Polynomial:

a-4z4 - a-3z3 + 4a-3z5 - a-2 + 3a-2z2 - 8a-2z4 + 9a-2z6 - a-1z + 7a-1z3 - 17a-1z5 + 13a-1z7 - z-2 + 1 + z2 + 5z4 - 19z6 + 13z8 + 2az-1 - 6az + 17az3 - 19az5 - 4az7 + 8az9 - 2a2z-2 + 7a2 - 20a2z2 + 48a2z4 - 54a2z6 + 16a2z8 + 2a2z10 + 2a3z-1 - 10a3z + 15a3z3 + 7a3z5 - 28a3z7 + 12a3z9 - a4z-2 + 9a4 - 27a4z2 + 48a4z4 - 37a4z6 + 6a4z8 + 2a4z10 - 7a5z + 11a5z3 + a5z5 - 10a5z7 + 4a5z9 + 3a6 - 9a6z2 + 14a6z4 - 11a6z6 + 3a6z8 - 2a7z + 5a7z3 - 4a7z5 + 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                 8 3     j = 1               11 6       j = -1             11 10         j = -3           10 9           j = -5         8 13             j = -7       5 8               j = -9     2 8                 j = -11   1 5                   j = -13   2                     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, 479]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 479]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 4, 11, 3], X[18, 7, 19, 8], X[22, 15, 17, 16], > X[20, 10, 21, 9], X[8, 13, 9, 14], X[14, 17, 15, 18], X[16, 21, 5, 22], > 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}, {7, -3, 9, -5, 8, -4}, > {10, -1, 3, -6, 5, -2, 11, -9, 6, -7, 4, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-7 3 7 13 16 21 20 2 3 4 19 - q + -- - -- + -- - -- + -- - -- - 14 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 -18 3 3 3 -10 9 3 6 3 2 4 -1 - q + q - --- + --- + --- + q + -- + -- + -- + -- + 4 q - 4 q + 16 14 12 8 6 4 2 q q q q q q q 6 8 10 12 > 2 q + 2 q - 2 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 479]][a, z]
Out[8]=	2 4 2 -2 2 4 6 -2 2 a a 2 z 4 2 6 2 -1 + a - a + 2 a - a + z - ---- + -- - 3 z + -- + 3 a z - a z - 2 2 2 z z a 4 4 z 2 4 4 4 6 2 6 > 2 z + -- - a z + 2 a z - z - a z 2 a
In[9]:=	Kauffman[Link[11, Alternating, 479]][a, z]
Out[9]=	2 4 3 -2 2 4 6 -2 2 a a 2 a 2 a z 1 - a + 7 a + 9 a + 3 a - z - ---- - -- + --- + ---- - - - 6 a z - 2 2 z z a z z 2 3 5 7 2 3 z 2 2 4 2 6 2 > 10 a z - 7 a z - 2 a z + z + ---- - 20 a z - 27 a z - 9 a z - 2 a 3 3 4 4 z 7 z 3 3 3 5 3 7 3 4 z 8 z > -- + ---- + 17 a z + 15 a z + 11 a z + 5 a z + 5 z + -- - ---- + 3 a 4 2 a a a 5 5 2 4 4 4 6 4 4 z 17 z 5 3 5 5 5 > 48 a z + 48 a z + 14 a z + ---- - ----- - 19 a z + 7 a z + a z - 3 a a 6 7 7 5 6 9 z 2 6 4 6 6 6 13 z 7 > 4 a z - 19 z + ---- - 54 a z - 37 a z - 11 a z + ----- - 4 a z - 2 a a 3 7 5 7 7 7 8 2 8 4 8 6 8 > 28 a z - 10 a z + a z + 13 z + 16 a z + 6 a z + 3 a z + 9 3 9 5 9 2 10 4 10 > 8 a z + 12 a z + 4 a z + 2 a z + 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	10 1 2 1 5 2 8 5 8 -- + 11 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 8 13 10 9 11 3 3 2 5 2 > ----- + ----- + ----- + ---- + --- + 6 q t + 8 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 L11a479

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