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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X14,7,15,8 X4,15,1,16 X5,10,6,11 X3849 X11,20,12,21 X17,5,18,22 X21,19,22,18 X19,12,20,13 X9,16,10,17 X13,2,14,3

Gauss Code:

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

Jones Polynomial:

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

A2 (sl(3)) Invariant:

- q-34 - q-32 - q-28 + q-26 + q-24 + 2q-22 + q-20 + 2q-16 + 2q-12 + q-10 + q-8 + q-2

HOMFLY-PT Polynomial:

- a3z-1 - 3a3z - 4a3z3 - a3z5 + 2a5z-1 + 5a5z + 7a5z3 + 5a5z5 + a5z7 - 3a7z-1 - 9a7z - 9a7z3 - 2a7z5 + 3a9z-1 + 5a9z + a9z3 - a11z-1

Kauffman Polynomial:

- a3z-1 + 4a3z - 7a3z3 + 5a3z5 - a3z7 + 4a4z2 - 13a4z4 + 10a4z6 - 2a4z8 - 2a5z-1 + 11a5z - 18a5z3 + 8a5z5 + 2a5z7 - a5z9 - 2a6 + 12a6z2 - 25a6z4 + 19a6z6 - 4a6z8 - 3a7z-1 + 17a7z - 25a7z3 + 12a7z5 + a7z7 - a7z9 + 4a8z2 - 9a8z4 + 8a8z6 - 2a8z8 - 3a9z-1 + 12a9z - 16a9z3 + 9a9z5 - 2a9z7 + 2a10 - 5a10z2 + 3a10z4 - a10z6 - a11z-1 + 2a11z - 2a11z3 + a12 - a12z2

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 0                   1 j = -2                 1   j = -4               2 1   j = -6           1 3 2     j = -8           3 1       j = -10       1 3 3         j = -12     1 4 3           j = -14     1 2             j = -16   2 3               j = -18   1                 j = -20 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, 1]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 1]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 7, 15, 8], X[4, 15, 1, 16], X[5, 10, 6, 11], > X[3, 8, 4, 9], X[11, 20, 12, 21], X[17, 5, 18, 22], X[21, 19, 22, 18], > X[19, 12, 20, 13], X[9, 16, 10, 17], X[13, 2, 14, 3]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 11, -5, -3}, {-4, -1, 2, 5, -10, 4, -6, 9, -11, -2, 3, 10, -7, 8, > -9, 6, -8, 7}]
In[6]:=	Jones[L][q]
Out[6]=	-(19/2) 2 3 4 4 5 3 3 2 1 q - ----- + ----- - ----- + ----- - ---- + ---- - ---- + ---- - ------- 17/2 15/2 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-34 -32 -28 -26 -24 2 -20 2 2 -10 -8 -2 -q - q - q + q + q + --- + q + --- + --- + q + q + q 22 16 12 q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 1]][a, z]
Out[8]=	3 5 7 9 11 a 2 a 3 a 3 a a 3 5 7 9 -(--) + ---- - ---- + ---- - --- - 3 a z + 5 a z - 9 a z + 5 a z - z z z z z 3 3 5 3 7 3 9 3 3 5 5 5 7 5 5 7 > 4 a z + 7 a z - 9 a z + a z - a z + 5 a z - 2 a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 1]][a, z]
Out[9]=	3 5 7 9 11 6 10 12 a 2 a 3 a 3 a a 3 5 -2 a + 2 a + a - -- - ---- - ---- - ---- - --- + 4 a z + 11 a z + z z z z z 7 9 11 4 2 6 2 8 2 10 2 > 17 a z + 12 a z + 2 a z + 4 a z + 12 a z + 4 a z - 5 a z - 12 2 3 3 5 3 7 3 9 3 11 3 4 4 > a z - 7 a z - 18 a z - 25 a z - 16 a z - 2 a z - 13 a z - 6 4 8 4 10 4 3 5 5 5 7 5 9 5 > 25 a z - 9 a z + 3 a z + 5 a z + 8 a z + 12 a z + 9 a z + 4 6 6 6 8 6 10 6 3 7 5 7 7 7 > 10 a z + 19 a z + 8 a z - a z - a z + 2 a z + a z - 9 7 4 8 6 8 8 8 5 9 7 9 > 2 a z - 2 a z - 4 a z - 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 1 1 2 3 1 1 2 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 6 4 20 7 18 6 16 6 16 5 14 5 12 5 14 4 q q q t q t q t q t q t q t q t 4 1 3 3 3 3 1 1 3 > ------ + ------ + ------ + ------ + ------ + ----- + ----- + ---- + ---- + 12 4 10 4 12 3 10 3 10 2 8 2 6 2 8 6 q t q t q t q t q t q t q t q t q t t t 2 > -- + -- + t 4 2 q q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n1

L11a548 L11n2