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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-21/2 + 2q-19/2 - 4q-17/2 + 6q-15/2 - 7q-13/2 + 8q-11/2 - 8q-9/2 + 6q-7/2 - 6q-5/2 + 3q-3/2 - 2q-1/2 + q1/2

A2 (sl(3)) Invariant:

q-30 + 2q-26 - q-20 - 2q-18 + 2q-16 + 4q-12 + 2q-10 + 2q-8 + q-6 - q-4 - 1

HOMFLY-PT Polynomial:

6a3z + 11a3z3 + 6a3z5 + a3z7 - a5z-1 - 18a5z - 33a5z3 - 24a5z5 - 8a5z7 - a5z9 + a7z-1 + 9a7z + 12a7z3 + 6a7z5 + a7z7

Kauffman Polynomial:

6a2z2 - 11a2z4 + 6a2z6 - a2z8 - 7a3z + 20a3z3 - 24a3z5 + 12a3z7 - 2a3z9 + 12a4z2 - 23a4z4 + 9a4z6 + 2a4z8 - a4z10 + a5z-1 - 18a5z + 40a5z3 - 46a5z5 + 26a5z7 - 5a5z9 - a6 + 16a6z2 - 29a6z4 + 18a6z6 - a6z8 - a6z10 + a7z-1 - 9a7z + 15a7z3 - 12a7z5 + 10a7z7 - 3a7z9 + 5a8z2 - 9a8z4 + 11a8z6 - 4a8z8 - 2a9z3 + 7a9z5 - 4a9z7 - 4a10z2 + 6a10z4 - 4a10z6 - a11z + 2a11z3 - 3a11z5 + a12z2 - 2a12z4 + a13z - a13z3

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 2                       1 j = 0                     1   j = -2                   2 1   j = -4                 4 1     j = -6               3 3       j = -8             5 3         j = -10           3 3           j = -12         4 5             j = -14       2 3               j = -16     2 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, Alternating, 374]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 374]]
Out[4]=	PD[X[12, 1, 13, 2], X[2, 13, 3, 14], X[14, 3, 15, 4], X[16, 5, 17, 6], > X[22, 7, 11, 8], X[18, 10, 19, 9], X[20, 18, 21, 17], X[10, 20, 1, 19], > X[8, 11, 9, 12], X[4, 15, 5, 16], X[6, 21, 7, 22]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -10, 4, -11, 5, -9, 6, -8}, > {9, -1, 2, -3, 10, -4, 7, -6, 8, -7, 11, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-(21/2) 2 4 6 7 8 8 6 6 3 -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 2 > ------- + Sqrt[q] Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-30 2 -20 2 2 4 2 2 -6 -4 -1 + q + --- - q - --- + --- + --- + --- + -- + q - q 26 18 16 12 10 8 q q q q q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 374]][a, z]
Out[8]=	5 7 a a 3 5 7 3 3 5 3 7 3 -(--) + -- + 6 a z - 18 a z + 9 a z + 11 a z - 33 a z + 12 a z + z z 3 5 5 5 7 5 3 7 5 7 7 7 5 9 > 6 a z - 24 a z + 6 a z + a z - 8 a z + a z - a z
In[9]:=	Kauffman[Link[11, Alternating, 374]][a, z]
Out[9]=	5 7 6 a a 3 5 7 11 13 2 2 -a + -- + -- - 7 a z - 18 a z - 9 a z - a z + a z + 6 a z + z z 4 2 6 2 8 2 10 2 12 2 3 3 5 3 > 12 a z + 16 a z + 5 a z - 4 a z + a z + 20 a z + 40 a z + 7 3 9 3 11 3 13 3 2 4 4 4 6 4 > 15 a z - 2 a z + 2 a z - a z - 11 a z - 23 a z - 29 a z - 8 4 10 4 12 4 3 5 5 5 7 5 9 5 > 9 a z + 6 a z - 2 a z - 24 a z - 46 a z - 12 a z + 7 a z - 11 5 2 6 4 6 6 6 8 6 10 6 3 7 > 3 a z + 6 a z + 9 a z + 18 a z + 11 a z - 4 a z + 12 a z + 5 7 7 7 9 7 2 8 4 8 6 8 8 8 > 26 a z + 10 a z - 4 a z - a z + 2 a z - a z - 4 a z - 3 9 5 9 7 9 4 10 6 10 > 2 a z - 5 a z - 3 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 4 1 1 1 3 2 4 2 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 6 4 22 8 20 7 18 7 18 6 16 6 16 5 14 5 q q q t q t q t q t q t q t q t 3 4 5 3 3 5 3 3 t > ------ + ------ + ------ + ------ + ------ + ----- + ---- + ---- + -- + 14 4 12 4 12 3 10 3 10 2 8 2 8 6 4 q t q t q t q t q t q t q t q t q 2 2 t 2 t 2 3 > --- + t + -- + q t 2 2 q q

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