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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X7,14,8,15 X17,20,18,21 X11,18,12,19 X19,12,20,13 X15,22,16,5 X21,16,22,17 X13,8,14,9 X2536 X4,9,1,10

Gauss Code:

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

Jones Polynomial:

q-25/2 - q-23/2 + q-21/2 - 2q-17/2 + 2q-15/2 - 3q-13/2 + 3q-11/2 - 3q-9/2 + q-7/2 - q-5/2

A2 (sl(3)) Invariant:

- q-40 - 2q-38 - q-36 - q-34 + q-32 + q-30 + q-28 + 2q-26 + 2q-22 + q-18 + 2q-16 + q-14 + 2q-12 + q-8

HOMFLY-PT Polynomial:

- a5z-1 - 4a5z - 4a5z3 - a5z5 + a7z-1 - 3a7z3 - a7z5 - a9z-1 + 2a11z-1 + 2a11z - a13z-1

Kauffman Polynomial:

a5z-1 - 4a5z + 4a5z3 - a5z5 - a6 + a6z2 + 2a6z4 - a6z6 + a7z-1 - 2a7z + a7z3 + 2a7z5 - a7z7 + 4a8z4 - 2a8z6 + a9z-1 - 3a9z + 4a9z3 - 2a9z5 - 4a10 + 16a10z2 - 20a10z4 + 8a10z6 - a10z8 + 2a11z-1 - 9a11z + 17a11z3 - 19a11z5 + 8a11z7 - a11z9 - 7a12 + 29a12z2 - 37a12z4 + 16a12z6 - 2a12z8 + a13z-1 - 4a13z + 10a13z3 - 14a13z5 + 7a13z7 - a13z9 - 3a14 + 12a14z2 - 15a14z4 + 7a14z6 - a14z8

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                     1 1 j = -8                   2     j = -10                 1 1     j = -12             1 3 2       j = -14             1 2         j = -16         1 3 2           j = -18       1   1             j = -20       1 2               j = -22   1 1                   j = -24                         j = -26 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, NonAlternating, 65]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 65]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[7, 14, 8, 15], X[17, 20, 18, 21], > X[11, 18, 12, 19], X[19, 12, 20, 13], X[15, 22, 16, 5], X[21, 16, 22, 17], > X[13, 8, 14, 9], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, -3, 9, 11, -2, -5, 6, -9, 3, -7, 8, -4, 5, > -6, 4, -8, 7}]
In[6]:=	Jones[L][q]
Out[6]=	-(25/2) -(23/2) -(21/2) 2 2 3 3 3 q - q + q - ----- + ----- - ----- + ----- - ---- + 17/2 15/2 13/2 11/2 9/2 q q q q q -(7/2) -(5/2) > q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-40 2 -36 -34 -32 -30 -28 2 2 -18 2 -q - --- - q - q + q + q + q + --- + --- + q + --- + 38 26 22 16 q q q q -14 2 -8 > q + --- + q 12 q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 65]][a, z]
Out[8]=	5 7 9 11 13 a a a 2 a a 5 11 5 3 7 3 5 5 -(--) + -- - -- + ----- - --- - 4 a z + 2 a z - 4 a z - 3 a z - a z - z z z z z 7 5 > a z
In[9]:=	Kauffman[Link[11, NonAlternating, 65]][a, z]
Out[9]=	5 7 9 11 13 6 10 12 14 a a a 2 a a 5 7 -a - 4 a - 7 a - 3 a + -- + -- + -- + ----- + --- - 4 a z - 2 a z - z z z z z 9 11 13 6 2 10 2 12 2 14 2 > 3 a z - 9 a z - 4 a z + a z + 16 a z + 29 a z + 12 a z + 5 3 7 3 9 3 11 3 13 3 6 4 8 4 > 4 a z + a z + 4 a z + 17 a z + 10 a z + 2 a z + 4 a z - 10 4 12 4 14 4 5 5 7 5 9 5 11 5 > 20 a z - 37 a z - 15 a z - a z + 2 a z - 2 a z - 19 a z - 13 5 6 6 8 6 10 6 12 6 14 6 7 7 > 14 a z - a z - 2 a z + 8 a z + 16 a z + 7 a z - a z + 11 7 13 7 10 8 12 8 14 8 11 9 13 9 > 8 a z + 7 a z - a z - 2 a z - a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-6 -4 1 1 1 1 1 2 1 q + q + ------- + ------- + ------ + ------ + ------ + ------ + ------ + 26 11 22 10 22 9 20 8 18 8 20 7 16 7 q t q t q t q t q t q t q t 1 3 2 1 1 2 3 2 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 18 6 16 6 16 5 14 5 12 5 14 4 12 4 12 3 q t q t q t q t q t q t q t q t 1 1 2 1 > ------ + ------ + ----- + ---- 10 3 10 2 8 2 6 q t q t q t q t

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