© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:

K11a291 K11a293

Knotscape

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DrawMorseLink

PD Presentation:

X6271 X10,4,11,3 X16,6,17,5 X22,8,1,7 X4,10,5,9 X18,12,19,11 X20,14,21,13 X2,16,3,15 X8,18,9,17 X14,20,15,19 X12,22,13,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 10 16 22 4 18 20 2 8 14 12

Alexander Polynomial:

10t-2 - 32t-1 + 45 - 32t + 10t2

Conway Polynomial:

1 + 8z2 + 10z4

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{129, 4}

Jones Polynomial:

q2 - 4q3 + 10q4 - 14q5 + 19q6 - 21q7 + 20q8 - 17q9 + 12q10 - 7q11 + 3q12 - q13

Other knots (up to mirrors) with the same Jones Polynomial:

{...}

A2 (sl(3)) Invariant:

q6 - 3q8 + 3q10 + 2q12 - 3q14 + 5q16 - q18 + q20 + 2q22 - 2q24 + 3q26 - 4q28 + 2q32 - 4q34 + q36 + q38 - q40

HOMFLY-PT Polynomial:

- a-12z2 - 2a-10 - 2a-10z2 + a-10z4 + a-8 + 5a-8z2 + 4a-8z4 + 2a-6 + 6a-6z2 + 4a-6z4 + a-4z4

Kauffman Polynomial:

- 2a-15z + 5a-15z3 - 4a-15z5 + a-15z7 - 5a-14z2 + 13a-14z4 - 11a-14z6 + 3a-14z8 - a-13z + 3a-13z3 + 6a-13z5 - 11a-13z7 + 4a-13z9 - 4a-12z2 + 24a-12z4 - 25a-12z6 + 4a-12z8 + 2a-12z10 - 3a-11z + 10a-11z3 + 3a-11z5 - 23a-11z7 + 11a-11z9 + 2a-10 - 9a-10z2 + 27a-10z4 - 41a-10z6 + 14a-10z8 + 2a-10z10 - 3a-9z + 18a-9z3 - 29a-9z5 + 3a-9z7 + 7a-9z9 + a-8 - 4a-8z2 + 5a-8z4 - 17a-8z6 + 13a-8z8 + a-7z + 6a-7z3 - 18a-7z5 + 14a-7z7 - 2a-6 + 6a-6z2 - 10a-6z4 + 10a-6z6 + 4a-5z5 + a-4z4

V2 and V3, the type 2 and 3 Vassiliev invariants:

{8, 22}

Khovanov Homology: (The squares with yellow highlighting are those on the "critical diagonals", where j-2r=s+1 or j-2r=s+1, where s=4 is the signature of 11292. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 r = 9 r = 10 r = 11 j = 27                       1 j = 25                     2   j = 23                   5 1   j = 21                 7 2     j = 19               10 5       j = 17             10 7         j = 15           11 10           j = 13         8 10             j = 11       6 11               j = 9     4 8                 j = 7     6                   j = 5 1 4                     j = 3 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]:=	Show[DrawMorseLink[Knot[11, Alternating, 292]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 292]]
Out[3]=	PD[X[6, 2, 7, 1], X[10, 4, 11, 3], X[16, 6, 17, 5], X[22, 8, 1, 7], > X[4, 10, 5, 9], X[18, 12, 19, 11], X[20, 14, 21, 13], X[2, 16, 3, 15], > X[8, 18, 9, 17], X[14, 20, 15, 19], X[12, 22, 13, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 292]]
Out[4]=	GaussCode[1, -8, 2, -5, 3, -1, 4, -9, 5, -2, 6, -11, 7, -10, 8, -3, 9, -6, 10, > -7, 11, -4]
In[5]:=	DTCode[Knot[11, Alternating, 292]]
Out[5]=	DTCode[6, 10, 16, 22, 4, 18, 20, 2, 8, 14, 12]
In[6]:=	alex = Alexander[Knot[11, Alternating, 292]][t]
Out[6]=	10 32 2 45 + -- - -- - 32 t + 10 t 2 t t
In[7]:=	Conway[Knot[11, Alternating, 292]][z]
Out[7]=	2 4 1 + 8 z + 10 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 292]}
In[9]:=	{KnotDet[Knot[11, Alternating, 292]], KnotSignature[Knot[11, Alternating, 292]]}
Out[9]=	{129, 4}
In[10]:=	J=Jones[Knot[11, Alternating, 292]][q]
Out[10]=	2 3 4 5 6 7 8 9 10 11 q - 4 q + 10 q - 14 q + 19 q - 21 q + 20 q - 17 q + 12 q - 7 q + 12 13 > 3 q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 292]}
In[12]:=	A2Invariant[Knot[11, Alternating, 292]][q]
Out[12]=	6 8 10 12 14 16 18 20 22 24 26 q - 3 q + 3 q + 2 q - 3 q + 5 q - q + q + 2 q - 2 q + 3 q - 28 32 34 36 38 40 > 4 q + 2 q - 4 q + q + q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 292]][a, z]
Out[13]=	2 2 2 2 4 4 4 4 -2 -8 2 z 2 z 5 z 6 z z 4 z 4 z z --- + a + -- - --- - ---- + ---- + ---- + --- + ---- + ---- + -- 10 6 12 10 8 6 10 8 6 4 a a a a a a a a a a
In[14]:=	Kauffman[Knot[11, Alternating, 292]][a, z]
Out[14]=	2 2 2 2 2 -8 2 2 z z 3 z 3 z z 5 z 4 z 9 z 4 z --- + a - -- - --- - --- - --- - --- + -- - ---- - ---- - ---- - ---- + 10 6 15 13 11 9 7 14 12 10 8 a a a a a a a a a a a 2 3 3 3 3 3 4 4 4 4 6 z 5 z 3 z 10 z 18 z 6 z 13 z 24 z 27 z 5 z > ---- + ---- + ---- + ----- + ----- + ---- + ----- + ----- + ----- + ---- - 6 15 13 11 9 7 14 12 10 8 a a a a a a a a a a 4 4 5 5 5 5 5 5 6 6 10 z z 4 z 6 z 3 z 29 z 18 z 4 z 11 z 25 z > ----- + -- - ---- + ---- + ---- - ----- - ----- + ---- - ----- - ----- - 6 4 15 13 11 9 7 5 14 12 a a a a a a a a a a 6 6 6 7 7 7 7 7 8 8 41 z 17 z 10 z z 11 z 23 z 3 z 14 z 3 z 4 z > ----- - ----- + ----- + --- - ----- - ----- + ---- + ----- + ---- + ---- + 10 8 6 15 13 11 9 7 14 12 a a a a a a a a a a 8 8 9 9 9 10 10 14 z 13 z 4 z 11 z 7 z 2 z 2 z > ----- + ----- + ---- + ----- + ---- + ----- + ----- 10 8 13 11 9 12 10 a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 292]], Vassiliev[3][Knot[11, Alternating, 292]]}
Out[15]=	{8, 22}
In[16]:=	Kh[Knot[11, Alternating, 292]][q, t]
Out[16]=	3 5 5 7 2 9 2 9 3 11 3 11 4 q + q + 4 q t + 6 q t + 4 q t + 8 q t + 6 q t + 11 q t + 13 4 13 5 15 5 15 6 17 6 17 7 > 8 q t + 10 q t + 11 q t + 10 q t + 10 q t + 7 q t + 19 7 19 8 21 8 21 9 23 9 23 10 > 10 q t + 5 q t + 7 q t + 2 q t + 5 q t + q t + 25 10 27 11 > 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a292

K11a291 K11a293