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DrawMorseLink

PD Presentation:

X10,1,11,2 X12,3,13,4 X14,5,15,6 X9,18,10,19 X17,22,18,9 X21,1,22,8 X20,15,21,16 X7,16,8,17 X6,20,7,19 X4,11,5,12 X2,13,3,14

Gauss Code:

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

Jones Polynomial:

q-21/2 - 2q-19/2 + 4q-17/2 - 7q-15/2 + 8q-13/2 - 9q-11/2 + 8q-9/2 - 7q-7/2 + 4q-5/2 - 2q-3/2

A2 (sl(3)) Invariant:

- q-32 - q-28 - 2q-26 + 2q-24 + 2q-20 + 3q-18 + q-16 + 3q-14 - q-12 + q-10 + q-8 - q-6 + 2q-4

HOMFLY-PT Polynomial:

- 3a3z - 2a3z3 - 2a5z-1 - 2a5z + a5z3 + a5z5 + 3a7z-1 + 5a7z + 3a7z3 + a7z5 - a9z-1 - 2a9z - a9z3

Kauffman Polynomial:

3a3z - 3a3z3 + a4z2 - 2a4z4 - a4z6 + 2a5z-1 - 5a5z + 2a5z3 - 2a5z7 - 3a6 + 6a6z2 - 5a6z4 + 2a6z6 - 2a6z8 + 3a7z-1 - 12a7z + 15a7z3 - 5a7z5 - a7z9 - 3a8 + 11a8z2 - 10a8z4 + 9a8z6 - 4a8z8 + a9z-1 - 2a9z + 3a9z3 + 2a9z5 - a9z9 - a10 + 2a10z2 - 3a10z4 + 5a10z6 - 2a10z8 + 2a11z - 7a11z3 + 7a11z5 - 2a11z7 - 4a12z2 + 4a12z4 - a12z6

Khovanov Homology:

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -2                   2 j = -4                 3 1 j = -6               4 1   j = -8             4 3     j = -10           5 4       j = -12         4 5         j = -14       3 4           j = -16     1 4             j = -18   1 3               j = -20   1                 j = -22 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, 215]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 215]]
Out[4]=	PD[X[10, 1, 11, 2], X[12, 3, 13, 4], X[14, 5, 15, 6], X[9, 18, 10, 19], > X[17, 22, 18, 9], X[21, 1, 22, 8], X[20, 15, 21, 16], X[7, 16, 8, 17], > X[6, 20, 7, 19], X[4, 11, 5, 12], X[2, 13, 3, 14]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 2, -10, 3, -9, -8, 6}, > {-4, -1, 10, -2, 11, -3, 7, 8, -5, 4, 9, -7, -6, 5}]
In[6]:=	Jones[L][q]
Out[6]=	-(21/2) 2 4 7 8 9 8 7 4 2 q - ----- + ----- - ----- + ----- - ----- + ---- - ---- + ---- - ---- 19/2 17/2 15/2 13/2 11/2 9/2 7/2 5/2 3/2 q q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-32 -28 2 2 2 3 -16 3 -12 -10 -8 -6 2 -q - q - --- + --- + --- + --- + q + --- - q + q + q - q + -- 26 24 20 18 14 4 q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 215]][a, z]
Out[8]=	5 7 9 -2 a 3 a a 3 5 7 9 3 3 5 3 ----- + ---- - -- - 3 a z - 2 a z + 5 a z - 2 a z - 2 a z + a z + z z z 7 3 9 3 5 5 7 5 > 3 a z - a z + a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 215]][a, z]
Out[9]=	5 7 9 6 8 10 2 a 3 a a 3 5 7 9 -3 a - 3 a - a + ---- + ---- + -- + 3 a z - 5 a z - 12 a z - 2 a z + z z z 11 4 2 6 2 8 2 10 2 12 2 3 3 > 2 a z + a z + 6 a z + 11 a z + 2 a z - 4 a z - 3 a z + 5 3 7 3 9 3 11 3 4 4 6 4 8 4 > 2 a z + 15 a z + 3 a z - 7 a z - 2 a z - 5 a z - 10 a z - 10 4 12 4 7 5 9 5 11 5 4 6 6 6 > 3 a z + 4 a z - 5 a z + 2 a z + 7 a z - a z + 2 a z + 8 6 10 6 12 6 5 7 11 7 6 8 8 8 > 9 a z + 5 a z - a z - 2 a z - 2 a z - 2 a z - 4 a z - 10 8 7 9 9 9 > 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-4 2 1 1 1 3 1 4 3 q + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 2 22 9 20 8 18 8 18 7 16 7 16 6 14 6 q q t q t q t q t q t q t q t 4 4 5 5 4 4 3 4 1 > ------ + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- + 14 5 12 5 12 4 10 4 10 3 8 3 8 2 6 2 6 q t q t q t q t q t q t q t q t q t 3 > ---- 4 q t

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