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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-11 + 3q-10 - 7q-9 + 13q-8 - 16q-7 + 20q-6 - 19q-5 + 17q-4 - 12q-3 + 8q-2 - 3q-1 + 1

A2 (sl(3)) Invariant:

- q-34 - q-32 + q-30 - 2q-28 + 4q-26 + 4q-24 + q-22 + 7q-20 + 5q-16 + 2q-14 + 5q-10 - 2q-8 + 3q-6 + q-4 - q-2 + 1

HOMFLY-PT Polynomial:

a2 + 2a2z2 + a2z4 + a4z-2 + 3a4 + 3a4z2 - a4z4 - a4z6 - 2a6z-2 - 8a6 - 10a6z2 - 7a6z4 - 2a6z6 + a8z-2 + 6a8 + 8a8z2 + 3a8z4 - 2a10 - a10z2

Kauffman Polynomial:

- a2 + 3a2z2 - 3a2z4 + a2z6 + 4a3z3 - 7a3z5 + 3a3z7 - a4z-2 + 5a4 - 7a4z2 + 8a4z4 - 11a4z6 + 5a4z8 + 2a5z-1 - 7a5z + 12a5z3 - 11a5z5 - 2a5z7 + 4a5z9 - 2a6z-2 + 14a6 - 36a6z2 + 48a6z4 - 40a6z6 + 13a6z8 + a6z10 + 2a7z-1 - 10a7z + 19a7z3 - 12a7z5 - 7a7z7 + 8a7z9 - a8z-2 + 10a8 - 31a8z2 + 46a8z4 - 38a8z6 + 14a8z8 + a8z10 - 3a9z + 14a9z3 - 15a9z5 + 3a9z7 + 4a9z9 - 2a10z2 + 4a10z4 - 7a10z6 + 6a10z8 + a11z + a11z3 - 6a11z5 + 5a11z7 - a12 + 3a12z2 - 5a12z4 + 3a12z6 + a13z - 2a13z3 + a13z5

Khovanov Homology:

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 1                       1 j = -1                     2   j = -3                   6 1   j = -5                 7 3     j = -7               10 5       j = -9             10 8         j = -11           10 9           j = -13         7 11             j = -15       6 9               j = -17     2 8                 j = -19   1 5                   j = -21   2                     j = -23 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]=	3
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 470]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 470]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[18, 11, 19, 12], X[8, 17, 9, 18], > X[16, 7, 17, 8], X[14, 15, 5, 16], X[22, 19, 15, 20], X[20, 14, 21, 13], > X[12, 22, 13, 21], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {6, -5, 4, -3, 7, -8, 9, -7}, > {10, -1, 5, -4, 11, -2, 3, -9, 8, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-11 3 7 13 16 20 19 17 12 8 3 1 - q + --- - -- + -- - -- + -- - -- + -- - -- + -- - - 10 9 8 7 6 5 4 3 2 q q q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-34 -32 -30 2 4 4 -22 7 5 2 5 2 1 - q - q + q - --- + --- + --- + q + --- + --- + --- + --- - -- + 28 26 24 20 16 14 10 8 q q q q q q q q 3 -4 -2 > -- + q - q 6 q
In[8]:=	HOMFLYPT[Link[11, Alternating, 470]][a, z]
Out[8]=	4 6 8 2 4 6 8 10 a 2 a a 2 2 4 2 a + 3 a - 8 a + 6 a - 2 a + -- - ---- + -- + 2 a z + 3 a z - 2 2 2 z z z 6 2 8 2 10 2 2 4 4 4 6 4 8 4 4 6 > 10 a z + 8 a z - a z + a z - a z - 7 a z + 3 a z - a z - 6 6 > 2 a z
In[9]:=	Kauffman[Link[11, Alternating, 470]][a, z]
Out[9]=	4 6 8 5 7 2 4 6 8 12 a 2 a a 2 a 2 a 5 -a + 5 a + 14 a + 10 a - a - -- - ---- - -- + ---- + ---- - 7 a z - 2 2 2 z z z z z 7 9 11 13 2 2 4 2 6 2 > 10 a z - 3 a z + a z + a z + 3 a z - 7 a z - 36 a z - 8 2 10 2 12 2 3 3 5 3 7 3 9 3 > 31 a z - 2 a z + 3 a z + 4 a z + 12 a z + 19 a z + 14 a z + 11 3 13 3 2 4 4 4 6 4 8 4 10 4 > a z - 2 a z - 3 a z + 8 a z + 48 a z + 46 a z + 4 a z - 12 4 3 5 5 5 7 5 9 5 11 5 13 5 > 5 a z - 7 a z - 11 a z - 12 a z - 15 a z - 6 a z + a z + 2 6 4 6 6 6 8 6 10 6 12 6 3 7 > a z - 11 a z - 40 a z - 38 a z - 7 a z + 3 a z + 3 a z - 5 7 7 7 9 7 11 7 4 8 6 8 8 8 > 2 a z - 7 a z + 3 a z + 5 a z + 5 a z + 13 a z + 14 a z + 10 8 5 9 7 9 9 9 6 10 8 10 > 6 a z + 4 a z + 8 a z + 4 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 6 1 2 1 5 2 8 6 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 5 3 23 9 21 8 19 8 19 7 17 7 17 6 15 6 q q q t q t q t q t q t q t q t 9 7 11 10 9 10 8 10 5 > ------ + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- + 15 5 13 5 13 4 11 4 11 3 9 3 9 2 7 2 7 q t q t q t q t q t q t q t q t q t 7 t 2 t 2 > ---- + -- + --- + q t 5 3 q q t q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a470

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