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

K11a212 K11a214

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X12,4,13,3 X16,6,17,5 X22,8,1,7 X18,10,19,9 X2,12,3,11 X20,13,21,14 X10,16,11,15 X6,18,7,17 X14,19,15,20 X8,22,9,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 12 16 22 18 2 20 10 6 14 8

Alexander Polynomial:

- 3t-3 + 16t-2 - 33t-1 + 41 - 33t + 16t2 - 3t3

Conway Polynomial:

1 + 4z2 - 2z4 - 3z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{145, 4}

Jones Polynomial:

1 - 3q + 8q2 - 14q3 + 20q4 - 23q5 + 24q6 - 21q7 + 16q8 - 10q9 + 4q10 - q11

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

{...}

A2 (sl(3)) Invariant:

1 - q2 + q4 + 3q6 - 4q8 + 4q10 - 2q12 - q14 + 4q16 - 3q18 + 5q20 - 3q22 + 2q26 - 4q28 + 2q30 - q34

HOMFLY-PT Polynomial:

- a-10 - a-10z2 + a-8 + 5a-8z2 + 3a-8z4 - 3a-6z2 - 5a-6z4 - 2a-6z6 + a-4z2 - a-4z4 - a-4z6 + a-2 + 2a-2z2 + a-2z4

Kauffman Polynomial:

- a-13z3 + a-13z5 + a-12z2 - 4a-12z4 + 4a-12z6 - 4a-11z + 10a-11z3 - 14a-11z5 + 9a-11z7 + a-10 - 4a-10z2 + 9a-10z4 - 16a-10z6 + 11a-10z8 - 5a-9z + 21a-9z3 - 25a-9z5 + 3a-9z7 + 7a-9z9 + a-8 - 7a-8z2 + 26a-8z4 - 39a-8z6 + 17a-8z8 + 2a-8z10 - 3a-7z + 12a-7z3 - 9a-7z5 - 13a-7z7 + 12a-7z9 - 5a-6z2 + 20a-6z4 - 29a-6z6 + 11a-6z8 + 2a-6z10 - 3a-5z + 7a-5z3 - 6a-5z5 - 4a-5z7 + 5a-5z9 + 4a-4z4 - 9a-4z6 + 5a-4z8 - a-3z + 5a-3z3 - 7a-3z5 + 3a-3z7 - a-2 + 3a-2z2 - 3a-2z4 + a-2z6

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

{4, 10}

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 11213. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 r = 9 j = 23                       1 j = 21                     3   j = 19                   7 1   j = 17                 9 3     j = 15               12 7       j = 13             12 9         j = 11           11 12           j = 9         9 12             j = 7       5 11               j = 5     3 9                 j = 3   1 6                   j = 1   2                     j = -1 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, 213]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 213]]
Out[3]=	PD[X[4, 2, 5, 1], X[12, 4, 13, 3], X[16, 6, 17, 5], X[22, 8, 1, 7], > X[18, 10, 19, 9], X[2, 12, 3, 11], X[20, 13, 21, 14], X[10, 16, 11, 15], > X[6, 18, 7, 17], X[14, 19, 15, 20], X[8, 22, 9, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 213]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -9, 4, -11, 5, -8, 6, -2, 7, -10, 8, -3, 9, -5, 10, > -7, 11, -4]
In[5]:=	DTCode[Knot[11, Alternating, 213]]
Out[5]=	DTCode[4, 12, 16, 22, 18, 2, 20, 10, 6, 14, 8]
In[6]:=	alex = Alexander[Knot[11, Alternating, 213]][t]
Out[6]=	3 16 33 2 3 41 - -- + -- - -- - 33 t + 16 t - 3 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 213]][z]
Out[7]=	2 4 6 1 + 4 z - 2 z - 3 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 213]}
In[9]:=	{KnotDet[Knot[11, Alternating, 213]], KnotSignature[Knot[11, Alternating, 213]]}
Out[9]=	{145, 4}
In[10]:=	J=Jones[Knot[11, Alternating, 213]][q]
Out[10]=	2 3 4 5 6 7 8 9 1 - 3 q + 8 q - 14 q + 20 q - 23 q + 24 q - 21 q + 16 q - 10 q + 10 11 > 4 q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 213]}
In[12]:=	A2Invariant[Knot[11, Alternating, 213]][q]
Out[12]=	2 4 6 8 10 12 14 16 18 20 1 - q + q + 3 q - 4 q + 4 q - 2 q - q + 4 q - 3 q + 5 q - 22 26 28 30 34 > 3 q + 2 q - 4 q + 2 q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 213]][a, z]
Out[13]=	2 2 2 2 2 4 4 4 4 -10 -8 -2 z 5 z 3 z z 2 z 3 z 5 z z z -a + a + a - --- + ---- - ---- + -- + ---- + ---- - ---- - -- + -- - 10 8 6 4 2 8 6 4 2 a a a a a a a a a 6 6 2 z z > ---- - -- 6 4 a a
In[14]:=	Kauffman[Knot[11, Alternating, 213]][a, z]
Out[14]=	2 2 2 2 -10 -8 -2 4 z 5 z 3 z 3 z z z 4 z 7 z 5 z a + a - a - --- - --- - --- - --- - -- + --- - ---- - ---- - ---- + 11 9 7 5 3 12 10 8 6 a a a a a a a a a 2 3 3 3 3 3 3 4 4 4 3 z z 10 z 21 z 12 z 7 z 5 z 4 z 9 z 26 z > ---- - --- + ----- + ----- + ----- + ---- + ---- - ---- + ---- + ----- + 2 13 11 9 7 5 3 12 10 8 a a a a a a a a a a 4 4 4 5 5 5 5 5 5 6 20 z 4 z 3 z z 14 z 25 z 9 z 6 z 7 z 4 z > ----- + ---- - ---- + --- - ----- - ----- - ---- - ---- - ---- + ---- - 6 4 2 13 11 9 7 5 3 12 a a a a a a a a a a 6 6 6 6 6 7 7 7 7 7 16 z 39 z 29 z 9 z z 9 z 3 z 13 z 4 z 3 z > ----- - ----- - ----- - ---- + -- + ---- + ---- - ----- - ---- + ---- + 10 8 6 4 2 11 9 7 5 3 a a a a a a a a a a 8 8 8 8 9 9 9 10 10 11 z 17 z 11 z 5 z 7 z 12 z 5 z 2 z 2 z > ----- + ----- + ----- + ---- + ---- + ----- + ---- + ----- + ----- 10 8 6 4 9 7 5 8 6 a a a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 213]], Vassiliev[3][Knot[11, Alternating, 213]]}
Out[15]=	{4, 10}
In[16]:=	Kh[Knot[11, Alternating, 213]][q, t]
Out[16]=	3 3 5 1 2 q q 5 7 7 2 9 2 6 q + 3 q + ---- + --- + -- + 9 q t + 5 q t + 11 q t + 9 q t + 2 t t q t 9 3 11 3 11 4 13 4 13 5 15 5 > 12 q t + 11 q t + 12 q t + 12 q t + 9 q t + 12 q t + 15 6 17 6 17 7 19 7 19 8 21 8 23 9 > 7 q t + 9 q t + 3 q t + 7 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a213

K11a212 K11a214