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L11n311 L11n313

Knotscape

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DrawMorseLink

PD Presentation:

X6172 X3,13,4,12 X9,20,10,21 X7,16,8,17 X13,18,14,19 X19,14,20,15 X15,22,16,11 X17,10,18,5 X21,8,22,9 X2536 X11,1,12,4

Gauss Code:

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

Jones Polynomial:

- 2q-9 + 5q-8 - 8q-7 + 11q-6 - 11q-5 + 12q-4 - 8q-3 + 7q-2 - 3q-1 + 1

A2 (sl(3)) Invariant:

- q-32 - q-30 - 3q-28 - q-26 + q-24 - q-22 + 5q-20 + 3q-18 + 6q-16 + 7q-14 + 3q-12 + 6q-10 + 2q-6 + q-4 - q-2 + 1

HOMFLY-PT Polynomial:

a2 + 2a2z2 + a2z4 + 2a4z-2 + 5a4 + 2a4z2 - 2a4z4 - a4z6 - 5a6z-2 - 12a6 - 9a6z2 - 4a6z4 - a6z6 + 4a8z-2 + 7a8 + 4a8z2 + a8z4 - a10z-2 - a10

Kauffman Polynomial:

- a2 + 3a2z2 - 3a2z4 + a2z6 + 4a3z3 - 8a3z5 + 3a3z7 - 2a4z-2 + 8a4 - 8a4z2 + 6a4z4 - 10a4z6 + 4a4z8 + 5a5z-1 - 18a5z + 25a5z3 - 20a5z5 + 2a5z7 + 2a5z9 - 5a6z-2 + 20a6 - 37a6z2 + 44a6z4 - 33a6z6 + 10a6z8 + 9a7z-1 - 33a7z + 46a7z3 - 27a7z5 + 4a7z7 + 2a7z9 - 4a8z-2 + 15a8 - 32a8z2 + 39a8z4 - 21a8z6 + 6a8z8 + 5a9z-1 - 19a9z + 28a9z3 - 15a9z5 + 5a9z7 - a10z-2 + 3a10 - 6a10z2 + 4a10z4 + a10z6 + a11z-1 - 4a11z + 3a11z3

Khovanov Homology:

trqj 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               5 1   j = -5             5 4     j = -7           7 3       j = -9         5 6         j = -11       6 6           j = -13     2 5             j = -15   3 6               j = -17   3                 j = -19 2

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, NonAlternating, 312]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 312]]
Out[4]=	PD[X[6, 1, 7, 2], X[3, 13, 4, 12], X[9, 20, 10, 21], X[7, 16, 8, 17], > X[13, 18, 14, 19], X[19, 14, 20, 15], X[15, 22, 16, 11], X[17, 10, 18, 5], > X[21, 8, 22, 9], X[2, 5, 3, 6], X[11, 1, 12, 4]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, -2, 11}, {10, -1, -4, 9, -3, 8}, > {-11, 2, -5, 6, -7, 4, -8, 5, -6, 3, -9, 7}]
In[6]:=	Jones[L][q]
Out[6]=	2 5 8 11 11 12 8 7 3 1 - -- + -- - -- + -- - -- + -- - -- + -- - - 9 8 7 6 5 4 3 2 q q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-32 -30 3 -26 -24 -22 5 3 6 7 3 1 - q - q - --- - q + q - q + --- + --- + --- + --- + --- + 28 20 18 16 14 12 q q q q q q 6 2 -4 -2 > --- + -- + q - q 10 6 q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 312]][a, z]
Out[8]=	4 6 8 10 2 4 6 8 10 2 a 5 a 4 a a 2 2 4 2 a + 5 a - 12 a + 7 a - a + ---- - ---- + ---- - --- + 2 a z + 2 a z - 2 2 2 2 z z z z 6 2 8 2 2 4 4 4 6 4 8 4 4 6 6 6 > 9 a z + 4 a z + a z - 2 a z - 4 a z + a z - a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 312]][a, z]
Out[9]=	4 6 8 10 5 7 2 4 6 8 10 2 a 5 a 4 a a 5 a 9 a -a + 8 a + 20 a + 15 a + 3 a - ---- - ---- - ---- - --- + ---- + ---- + 2 2 2 2 z z z z z z 9 11 5 a a 5 7 9 11 2 2 4 2 > ---- + --- - 18 a z - 33 a z - 19 a z - 4 a z + 3 a z - 8 a z - z z 6 2 8 2 10 2 3 3 5 3 7 3 9 3 > 37 a z - 32 a z - 6 a z + 4 a z + 25 a z + 46 a z + 28 a z + 11 3 2 4 4 4 6 4 8 4 10 4 3 5 > 3 a z - 3 a z + 6 a z + 44 a z + 39 a z + 4 a z - 8 a z - 5 5 7 5 9 5 2 6 4 6 6 6 8 6 > 20 a z - 27 a z - 15 a z + a z - 10 a z - 33 a z - 21 a z + 10 6 3 7 5 7 7 7 9 7 4 8 6 8 > a z + 3 a z + 2 a z + 4 a z + 5 a z + 4 a z + 10 a z + 8 8 5 9 7 9 > 6 a z + 2 a z + 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	4 5 2 3 3 6 2 5 6 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 5 3 19 7 17 6 15 6 15 5 13 5 13 4 11 4 q q q t q t q t q t q t q t q t 6 5 6 7 3 5 t 2 t 2 > ------ + ----- + ----- + ----- + ---- + ---- + -- + --- + q t 11 3 9 3 9 2 7 2 7 5 3 q q t q t q t q t q t q t q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n312

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