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DrawMorseLink

PD Presentation:

X10,1,11,2 X2,11,3,12 X12,3,13,4 X14,5,15,6 X6,9,7,10 X18,8,19,7 X22,16,9,15 X20,18,21,17 X16,22,17,21 X8,20,1,19 X4,13,5,14

Gauss Code:

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

Jones Polynomial:

- q-15/2 + 2q-13/2 - 4q-11/2 + 6q-9/2 - 7q-7/2 + 7q-5/2 - 8q-3/2 + 6q-1/2 - 5q1/2 + 3q3/2 - 2q5/2 + q7/2

A2 (sl(3)) Invariant:

q-22 + q-18 + q-16 + q-12 - q-10 + 2q-8 + q-6 + q-4 + 2q-2 + q2 - q10

HOMFLY-PT Polynomial:

3a-1z + 4a-1z3 + a-1z5 - 3az - 7az3 - 5az5 - az7 - a3z-1 - 6a3z - 8a3z3 - 5a3z5 - a3z7 + a5z-1 + 4a5z + 4a5z3 + a5z5

Kauffman Polynomial:

6a-2z2 - 11a-2z4 + 6a-2z6 - a-2z8 - 4a-1z + 16a-1z3 - 23a-1z5 + 12a-1z7 - 2a-1z9 + 6z2 - 12z4 + 3z6 + 3z8 - z10 - 3az + 14az3 - 27az5 + 19az7 - 4az9 + a2z2 - 6a2z4 + 6a2z6 + a2z8 - a2z10 - a3z-1 + 7a3z - 11a3z3 + 7a3z5 + 3a3z7 - 2a3z9 + a4 - 4a4z2 + 3a4z4 + 5a4z6 - 3a4z8 - a5z-1 + 4a5z - 6a5z3 + 8a5z5 - 4a5z7 - 4a6z2 + 6a6z4 - 4a6z6 - a7z + 2a7z3 - 3a7z5 + a8z2 - 2a8z4 + a9z - a9z3

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 8                       1 j = 6                     1   j = 4                   2 1   j = 2                 3 1     j = 0               3 2       j = -2             5 3         j = -4           3 4           j = -6         4 4             j = -8       2 3               j = -10     2 4                 j = -12   1 3                   j = -14   1                     j = -16 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, Alternating, 275]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 275]]
Out[4]=	PD[X[10, 1, 11, 2], X[2, 11, 3, 12], X[12, 3, 13, 4], X[14, 5, 15, 6], > X[6, 9, 7, 10], X[18, 8, 19, 7], X[22, 16, 9, 15], X[20, 18, 21, 17], > X[16, 22, 17, 21], X[8, 20, 1, 19], X[4, 13, 5, 14]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -11, 4, -5, 6, -10}, > {5, -1, 2, -3, 11, -4, 7, -9, 8, -6, 10, -8, 9, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-(15/2) 2 4 6 7 7 8 6 -q + ----- - ----- + ---- - ---- + ---- - ---- + ------- - 5 Sqrt[q] + 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q 3/2 5/2 7/2 > 3 q - 2 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 -18 -16 -12 -10 2 -6 -4 2 2 10 q + q + q + q - q + -- + q + q + -- + q - q 8 2 q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 275]][a, z]
Out[8]=	3 5 3 a a 3 z 3 5 4 z 3 3 3 -(--) + -- + --- - 3 a z - 6 a z + 4 a z + ---- - 7 a z - 8 a z + z z a a 5 5 3 z 5 3 5 5 5 7 3 7 > 4 a z + -- - 5 a z - 5 a z + a z - a z - a z a
In[9]:=	Kauffman[Link[11, Alternating, 275]][a, z]
Out[9]=	3 5 2 4 a a 4 z 3 5 7 9 2 6 z a - -- - -- - --- - 3 a z + 7 a z + 4 a z - a z + a z + 6 z + ---- + z z a 2 a 3 2 2 4 2 6 2 8 2 16 z 3 3 3 5 3 > a z - 4 a z - 4 a z + a z + ----- + 14 a z - 11 a z - 6 a z + a 4 7 3 9 3 4 11 z 2 4 4 4 6 4 8 4 > 2 a z - a z - 12 z - ----- - 6 a z + 3 a z + 6 a z - 2 a z - 2 a 5 6 23 z 5 3 5 5 5 7 5 6 6 z 2 6 > ----- - 27 a z + 7 a z + 8 a z - 3 a z + 3 z + ---- + 6 a z + a 2 a 7 8 4 6 6 6 12 z 7 3 7 5 7 8 z > 5 a z - 4 a z + ----- + 19 a z + 3 a z - 4 a z + 3 z - -- + a 2 a 9 2 8 4 8 2 z 9 3 9 10 2 10 > a z - 3 a z - ---- - 4 a z - 2 a z - z - a z a
In[10]:=	Kh[L][q, t]
Out[10]=	4 5 1 1 1 3 2 4 2 3 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 4 2 16 6 14 5 12 5 12 4 10 4 10 3 8 3 8 2 q q q t q t q t q t q t q t q t q t 4 4 3 3 t 2 2 2 2 3 4 3 > ----- + ---- + ---- + 3 t + --- + 2 t + 3 q t + q t + 2 q t + 6 2 6 4 2 q t q t q t q 4 4 6 4 8 5 > q t + q t + q t

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