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

K11a213 K11a215

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 12 16 22 20 18 2 6 14 10 8

Alexander Polynomial:

3t-2 - 17t-1 + 29 - 17t + 3t2

Conway Polynomial:

1 - 5z2 + 3z4

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{69, 0}

Jones Polynomial:

q-4 - 2q-3 + 5q-2 - 7q-1 + 9 - 11q + 10q2 - 9q3 + 7q4 - 4q5 + 3q6 - q7

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

{...}

A2 (sl(3)) Invariant:

q-14 + q-12 - q-10 + 2q-8 + q-6 - 2q-4 + 2q-2 - 1 - q2 - q4 - 2q6 + q8 - q10 + q12 + 3q14 - q16 + q18 + q20 - q22

HOMFLY-PT Polynomial:

- a-6z2 + 2a-4 + a-4z2 + a-4z4 - 2a-2 - 2a-2z2 + a-2z4 - z2 + z4 - 2a2z2 + a4

Kauffman Polynomial:

3a-7z3 - 4a-7z5 + a-7z7 - 7a-6z2 + 18a-6z4 - 14a-6z6 + 3a-6z8 + 3a-5z - 5a-5z3 + 12a-5z5 - 12a-5z7 + 3a-5z9 + 2a-4 - 24a-4z2 + 45a-4z4 - 28a-4z6 + 3a-4z8 + a-4z10 + 11a-3z - 29a-3z3 + 37a-3z5 - 25a-3z7 + 6a-3z9 + 2a-2 - 21a-2z2 + 34a-2z4 - 21a-2z6 + 3a-2z8 + a-2z10 + 8a-1z - 19a-1z3 + 17a-1z5 - 9a-1z7 + 3a-1z9 - z2 + 3z4 - 4z6 + 3z8 - 2az5 + 3az7 + a2z2 - 3a2z4 + 3a2z6 - 2a3z3 + 2a3z5 + a4 - 2a4z2 + a4z4

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

{-5, -1}

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=0 is the signature of 11214. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 15                       1 j = 13                     2   j = 11                   2 1   j = 9                 5 2     j = 7               4 2       j = 5             6 5         j = 3           5 4           j = 1         4 6             j = -1       4 6               j = -3     1 3                 j = -5   1 4                   j = -7   1                     j = -9 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, 214]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 214]]
Out[3]=	PD[X[4, 2, 5, 1], X[12, 3, 13, 4], X[16, 5, 17, 6], X[22, 8, 1, 7], > X[20, 10, 21, 9], X[18, 12, 19, 11], X[2, 13, 3, 14], X[6, 15, 7, 16], > X[14, 18, 15, 17], X[10, 20, 11, 19], X[8, 22, 9, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 214]]
Out[4]=	GaussCode[1, -7, 2, -1, 3, -8, 4, -11, 5, -10, 6, -2, 7, -9, 8, -3, 9, -6, 10, > -5, 11, -4]
In[5]:=	DTCode[Knot[11, Alternating, 214]]
Out[5]=	DTCode[4, 12, 16, 22, 20, 18, 2, 6, 14, 10, 8]
In[6]:=	alex = Alexander[Knot[11, Alternating, 214]][t]
Out[6]=	3 17 2 29 + -- - -- - 17 t + 3 t 2 t t
In[7]:=	Conway[Knot[11, Alternating, 214]][z]
Out[7]=	2 4 1 - 5 z + 3 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 214]}
In[9]:=	{KnotDet[Knot[11, Alternating, 214]], KnotSignature[Knot[11, Alternating, 214]]}
Out[9]=	{69, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 214]][q]
Out[10]=	-4 2 5 7 2 3 4 5 6 7 9 + q - -- + -- - - - 11 q + 10 q - 9 q + 7 q - 4 q + 3 q - q 3 2 q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 214]}
In[12]:=	A2Invariant[Knot[11, Alternating, 214]][q]
Out[12]=	-14 -12 -10 2 -6 2 2 2 4 6 8 10 -1 + q + q - q + -- + q - -- + -- - q - q - 2 q + q - q + 8 4 2 q q q 12 14 16 18 20 22 > q + 3 q - q + q + q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 214]][a, z]
Out[13]=	2 2 2 4 4 2 2 4 2 z z 2 z 2 2 4 z z -- - -- + a - z - -- + -- - ---- - 2 a z + z + -- + -- 4 2 6 4 2 4 2 a a a a a a a
In[14]:=	Kauffman[Knot[11, Alternating, 214]][a, z]
Out[14]=	2 2 2 2 2 4 3 z 11 z 8 z 2 7 z 24 z 21 z 2 2 4 2 -- + -- + a + --- + ---- + --- - z - ---- - ----- - ----- + a z - 2 a z + 4 2 5 3 a 6 4 2 a a a a a a a 3 3 3 3 4 4 4 3 z 5 z 29 z 19 z 3 3 4 18 z 45 z 34 z > ---- - ---- - ----- - ----- - 2 a z + 3 z + ----- + ----- + ----- - 7 5 3 a 6 4 2 a a a a a a 5 5 5 5 2 4 4 4 4 z 12 z 37 z 17 z 5 3 5 6 > 3 a z + a z - ---- + ----- + ----- + ----- - 2 a z + 2 a z - 4 z - 7 5 3 a a a a 6 6 6 7 7 7 7 14 z 28 z 21 z 2 6 z 12 z 25 z 9 z 7 > ----- - ----- - ----- + 3 a z + -- - ----- - ----- - ---- + 3 a z + 6 4 2 7 5 3 a a a a a a a 8 8 8 9 9 9 10 10 8 3 z 3 z 3 z 3 z 6 z 3 z z z > 3 z + ---- + ---- + ---- + ---- + ---- + ---- + --- + --- 6 4 2 5 3 a 4 2 a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 214]], Vassiliev[3][Knot[11, Alternating, 214]]}
Out[15]=	{-5, -1}
In[16]:=	Kh[Knot[11, Alternating, 214]][q, t]
Out[16]=	6 1 1 1 4 1 3 4 3 - + 4 q + ----- + ----- + ----- + ----- + ----- + ---- + --- + 6 q t + 5 q t + q 9 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t 3 2 5 2 5 3 7 3 7 4 9 4 9 5 > 4 q t + 6 q t + 5 q t + 4 q t + 2 q t + 5 q t + 2 q t + 11 5 11 6 13 6 15 7 > 2 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a214

K11a213 K11a215