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DrawMorseLink

PD Presentation:

X8192 X2,9,3,10 X10,3,11,4 X14,5,15,6 X22,17,13,18 X20,15,21,16 X16,21,17,22 X12,19,7,20 X18,11,19,12 X6718 X4,13,5,14

Gauss Code:

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

Jones Polynomial:

- q-15 + 3q-14 - 6q-13 + 9q-12 - 11q-11 + 12q-10 - 11q-9 + 10q-8 - 6q-7 + 5q-6 - q-5 + q-4

A2 (sl(3)) Invariant:

- 2q-44 - 2q-40 - q-34 + 4q-32 + 6q-28 + 4q-26 + 5q-24 + 6q-22 + 2q-20 + 4q-18 + q-14

HOMFLY-PT Polynomial:

2a8z-2 + 11a8 + 21a8z2 + 18a8z4 + 7a8z6 + a8z8 - 5a10z-2 - 19a10 - 18a10z2 - a10z4 + 4a10z6 + a10z8 + 4a12z-2 + 9a12 + 2a12z2 - 3a12z4 - a12z6 - a14z-2 - a14

Kauffman Polynomial:

- 2a8z-2 + 11a8 - 21a8z2 + 18a8z4 - 7a8z6 + a8z8 + 5a9z-1 - 19a9z + 20a9z3 - 4a9z5 - 3a9z7 + a9z9 - 5a10z-2 + 22a10 - 39a10z2 + 36a10z4 - 13a10z6 - a10z8 + a10z10 + 9a11z-1 - 35a11z + 39a11z3 - a11z5 - 16a11z7 + 5a11z9 - 4a12z-2 + 13a12 - 20a12z2 + 37a12z4 - 32a12z6 + 6a12z8 + a12z10 + 5a13z-1 - 19a13z + 33a13z3 - 23a13z5 - 3a13z7 + 4a13z9 - a14z-2 + 3a14z2 + 3a14z4 - 17a14z6 + 8a14z8 + a15z-1 - 2a15z + 9a15z3 - 20a15z5 + 10a15z7 - a16 + 5a16z2 - 13a16z4 + 9a16z6 + a17z - 4a17z3 + 6a17z5 + 3a18z4 + a19z3

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 = -7                       1 j = -9                     1 1 j = -11                   4     j = -13                 2 1     j = -15               8 4       j = -17             4 3         j = -19           8 7           j = -21         5 6             j = -23       4 6               j = -25     2 5                 j = -27   1 4                   j = -29   2                     j = -31 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, 504]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 504]]
Out[4]=	PD[X[8, 1, 9, 2], X[2, 9, 3, 10], X[10, 3, 11, 4], X[14, 5, 15, 6], > X[22, 17, 13, 18], X[20, 15, 21, 16], X[16, 21, 17, 22], X[12, 19, 7, 20], > X[18, 11, 19, 12], X[6, 7, 1, 8], X[4, 13, 5, 14]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -11, 4, -10}, {10, -1, 2, -3, 9, -8}, > {11, -4, 6, -7, 5, -9, 8, -6, 7, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-15 3 6 9 11 12 11 10 6 5 -5 -4 -q + --- - --- + --- - --- + --- - -- + -- - -- + -- - q + q 14 13 12 11 10 9 8 7 6 q q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-2 2 -34 4 6 4 5 6 2 4 -14 --- - --- - q + --- + --- + --- + --- + --- + --- + --- + q 44 40 32 28 26 24 22 20 18 q q q q q q q q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 504]][a, z]
Out[8]=	8 10 12 14 8 10 12 14 2 a 5 a 4 a a 8 2 11 a - 19 a + 9 a - a + ---- - ----- + ----- - --- + 21 a z - 2 2 2 2 z z z z 10 2 12 2 8 4 10 4 12 4 8 6 10 6 > 18 a z + 2 a z + 18 a z - a z - 3 a z + 7 a z + 4 a z - 12 6 8 8 10 8 > a z + a z + a z
In[9]:=	Kauffman[Link[11, Alternating, 504]][a, z]
Out[9]=	8 10 12 14 9 11 8 10 12 16 2 a 5 a 4 a a 5 a 9 a 11 a + 22 a + 13 a - a - ---- - ----- - ----- - --- + ---- + ----- + 2 2 2 2 z z z z z z 13 15 5 a a 9 11 13 15 17 8 2 > ----- + --- - 19 a z - 35 a z - 19 a z - 2 a z + a z - 21 a z - z z 10 2 12 2 14 2 16 2 9 3 11 3 > 39 a z - 20 a z + 3 a z + 5 a z + 20 a z + 39 a z + 13 3 15 3 17 3 19 3 8 4 10 4 > 33 a z + 9 a z - 4 a z + a z + 18 a z + 36 a z + 12 4 14 4 16 4 18 4 9 5 11 5 > 37 a z + 3 a z - 13 a z + 3 a z - 4 a z - a z - 13 5 15 5 17 5 8 6 10 6 12 6 > 23 a z - 20 a z + 6 a z - 7 a z - 13 a z - 32 a z - 14 6 16 6 9 7 11 7 13 7 15 7 8 8 > 17 a z + 9 a z - 3 a z - 16 a z - 3 a z + 10 a z + a z - 10 8 12 8 14 8 9 9 11 9 13 9 10 10 > a z + 6 a z + 8 a z + a z + 5 a z + 4 a z + a z + 12 10 > a z
In[10]:=	Kh[L][q, t]
Out[10]=	-9 -7 1 2 1 4 2 5 4 q + q + ------- + ------- + ------- + ------ + ------ + ------ + ------ + 31 11 29 10 27 10 27 9 25 9 25 8 23 8 q t q t q t q t q t q t q t 6 5 6 8 7 4 3 8 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 23 7 21 7 21 6 19 6 19 5 17 5 17 4 15 4 q t q t q t q t q t q t q t q t 4 2 1 4 1 > ------ + ------ + ------ + ------ + ---- 15 3 13 3 13 2 11 2 9 q t q t q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a504

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