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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-27/2 - 2q-25/2 + 6q-23/2 - 11q-21/2 + 16q-19/2 - 22q-17/2 + 19q-15/2 - 21q-13/2 + 14q-11/2 - 11q-9/2 + 4q-7/2 - q-5/2

A2 (sl(3)) Invariant:

- q-44 - 4q-42 - 5q-40 - 4q-38 - 6q-36 + 9q-32 + 8q-30 + 18q-28 + 17q-26 + 15q-24 + 16q-22 + 5q-20 + 10q-18 + q-16 + 4q-12 - 3q-10 + q-8

HOMFLY-PT Polynomial:

- a5z3 - a5z5 - 2a7z-3 - 9a7z-1 - 16a7z - 13a7z3 - 4a7z5 + 7a9z-3 + 23a9z-1 + 26a9z + 10a9z3 - 9a11z-3 - 19a11z-1 - 10a11z + 5a13z-3 + 5a13z-1 - a15z-3

Kauffman Polynomial:

a5z3 - a5z5 + 3a6z4 - 4a6z6 + 2a7z-3 - 9a7z-1 + 16a7z - 19a7z3 + 19a7z5 - 10a7z7 - 7a8z-2 + 23a8 - 26a8z2 + 12a8z4 + 10a8z6 - 10a8z8 + 7a9z-3 - 23a9z-1 + 39a9z - 47a9z3 + 40a9z5 - 10a9z7 - 5a9z9 - 19a10z-2 + 60a10 - 73a10z2 + 29a10z4 + 19a10z6 - 15a10z8 - a10z10 + 9a11z-3 - 24a11z-1 + 37a11z - 45a11z3 + 34a11z5 - 3a11z7 - 7a11z9 - 18a12z-2 + 58a12 - 75a12z2 + 38a12z4 + 3a12z6 - 7a12z8 - a12z10 + 5a13z-3 - 12a13z-1 + 16a13z - 22a13z3 + 19a13z5 - 5a13z7 - 2a13z9 - 7a14z-2 + 24a14 - 34a14z2 + 22a14z4 - 3a14z6 - 2a14z8 + a15z-3 - 2a15z-1 + 2a15z - 4a15z3 + 5a15z5 - 2a15z7 - a16z-2 + 4a16 - 6a16z2 + 4a16z4 - a16z6

Khovanov Homology:

trqj r = -11 r = -10 r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -4                       1 j = -6                     4 1 j = -8                   7     j = -10                 7 4     j = -12               14 7       j = -14             11 13         j = -16           11 8           j = -18         5 11             j = -20       6 11               j = -22     1 6                 j = -24   1 5                   j = -26   1                     j = -28 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]=	4
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 531]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 531]]
Out[4]=	PD[X[6, 1, 7, 2], X[2, 5, 3, 6], X[18, 11, 19, 12], X[10, 3, 11, 4], > X[4, 9, 1, 10], X[14, 7, 15, 8], X[8, 13, 5, 14], X[22, 19, 13, 20], > X[20, 15, 21, 16], X[16, 21, 17, 22], X[12, 17, 9, 18]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 4, -5}, {2, -1, 6, -7}, {5, -4, 3, -11}, > {7, -6, 9, -10, 11, -3, 8, -9, 10, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-(27/2) 2 6 11 16 22 19 21 14 q - ----- + ----- - ----- + ----- - ----- + ----- - ----- + ----- - 25/2 23/2 21/2 19/2 17/2 15/2 13/2 11/2 q q q q q q q q 11 4 -(5/2) > ---- + ---- - q 9/2 7/2 q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-44 4 5 4 6 9 8 18 17 15 16 5 10 -q - --- - --- - --- - --- + --- + --- + --- + --- + --- + --- + --- + --- + 42 40 38 36 32 30 28 26 24 22 20 18 q q q q q q q q q q q q -16 4 3 -8 > q + --- - --- + q 12 10 q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 531]][a, z]
Out[8]=	7 9 11 13 15 7 9 11 13 -2 a 7 a 9 a 5 a a 9 a 23 a 19 a 5 a 7 ----- + ---- - ----- + ----- - --- - ---- + ----- - ------ + ----- - 16 a z + 3 3 3 3 3 z z z z z z z z z 9 11 5 3 7 3 9 3 5 5 7 5 > 26 a z - 10 a z - a z - 13 a z + 10 a z - a z - 4 a z
In[9]:=	Kauffman[Link[11, Alternating, 531]][a, z]
Out[9]=	7 9 11 13 15 8 10 12 14 16 2 a 7 a 9 a 5 a a 23 a + 60 a + 58 a + 24 a + 4 a + ---- + ---- + ----- + ----- + --- - 3 3 3 3 3 z z z z z 8 10 12 14 16 7 9 11 13 7 a 19 a 18 a 7 a a 9 a 23 a 24 a 12 a > ---- - ------ - ------ - ----- - --- - ---- - ----- - ------ - ------ - 2 2 2 2 2 z z z z z z z z z 15 2 a 7 9 11 13 15 8 2 > ----- + 16 a z + 39 a z + 37 a z + 16 a z + 2 a z - 26 a z - z 10 2 12 2 14 2 16 2 5 3 7 3 > 73 a z - 75 a z - 34 a z - 6 a z + a z - 19 a z - 9 3 11 3 13 3 15 3 6 4 8 4 > 47 a z - 45 a z - 22 a z - 4 a z + 3 a z + 12 a z + 10 4 12 4 14 4 16 4 5 5 7 5 > 29 a z + 38 a z + 22 a z + 4 a z - a z + 19 a z + 9 5 11 5 13 5 15 5 6 6 8 6 > 40 a z + 34 a z + 19 a z + 5 a z - 4 a z + 10 a z + 10 6 12 6 14 6 16 6 7 7 9 7 11 7 > 19 a z + 3 a z - 3 a z - a z - 10 a z - 10 a z - 3 a z - 13 7 15 7 8 8 10 8 12 8 14 8 > 5 a z - 2 a z - 10 a z - 15 a z - 7 a z - 2 a z - 9 9 11 9 13 9 10 10 12 10 > 5 a z - 7 a z - 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-6 -4 1 1 1 5 1 6 6 q + q + ------- + ------- + ------- + ------ + ------ + ------ + ------ + 28 11 26 10 24 10 24 9 22 9 22 8 20 8 q t q t q t q t q t q t q t 11 5 11 11 8 11 13 14 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 20 7 18 7 18 6 16 6 16 5 14 5 14 4 12 4 q t q t q t q t q t q t q t q t 7 7 4 7 4 > ------ + ------ + ------ + ----- + ---- 12 3 10 3 10 2 8 2 6 q t q t q t q t q t

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