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

K11a214 K11a216

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 12 18 14 16 2 22 20 10 6 8

Alexander Polynomial:

t-4 - 6t-3 + 17t-2 - 27t-1 + 31 - 27t + 17t2 - 6t3 + t4

Conway Polynomial:

1 + 3z2 + z4 + 2z6 + z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{133, 4}

Jones Polynomial:

- q-1 + 4 - 8q + 14q2 - 18q3 + 21q4 - 21q5 + 19q6 - 14q7 + 8q8 - 4q9 + q10

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

{...}

A2 (sl(3)) Invariant:

- q-2 + 2 - 2q2 + 2q4 + 2q6 - 2q8 + 5q10 - 4q12 + 3q14 - q18 + 3q20 - 4q22 + q24 - q26 - q28 + q30

HOMFLY-PT Polynomial:

2a-8z2 + a-8z4 - 2a-6 - 7a-6z2 - 7a-6z4 - 2a-6z6 + 3a-4 + 10a-4z2 + 10a-4z4 + 5a-4z6 + a-4z8 - 2a-2z2 - 3a-2z4 - a-2z6

Kauffman Polynomial:

a-12z4 - 2a-11z3 + 4a-11z5 + a-10z2 - 6a-10z4 + 8a-10z6 + 6a-9z3 - 15a-9z5 + 12a-9z7 - 2a-8z2 + 11a-8z4 - 21a-8z6 + 13a-8z8 - 5a-7z + 23a-7z3 - 26a-7z5 - 2a-7z7 + 8a-7z9 + 2a-6 - 18a-6z2 + 56a-6z4 - 64a-6z6 + 19a-6z8 + 2a-6z10 - 8a-5z + 22a-5z3 - 3a-5z5 - 27a-5z7 + 13a-5z9 + 3a-4 - 21a-4z2 + 54a-4z4 - 49a-4z6 + 10a-4z8 + 2a-4z10 - 4a-3z + 10a-3z3 + a-3z5 - 12a-3z7 + 5a-3z9 - 6a-2z2 + 16a-2z4 - 14a-2z6 + 4a-2z8 - a-1z + 3a-1z3 - 3a-1z5 + a-1z7

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

{3, 5}

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 11215. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 j = 21                       1 j = 19                     3   j = 17                   5 1   j = 15                 9 3     j = 13               10 5       j = 11             11 9         j = 9           10 10           j = 7         8 11             j = 5       6 10               j = 3     3 9                 j = 1   1 5                   j = -1   3                     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, 215]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 215]]
Out[3]=	PD[X[4, 2, 5, 1], X[12, 4, 13, 3], X[18, 6, 19, 5], X[14, 7, 15, 8], > X[16, 10, 17, 9], X[2, 12, 3, 11], X[22, 13, 1, 14], X[20, 16, 21, 15], > X[10, 18, 11, 17], X[6, 20, 7, 19], X[8, 21, 9, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 215]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -10, 4, -11, 5, -9, 6, -2, 7, -4, 8, -5, 9, -3, 10, > -8, 11, -7]
In[5]:=	DTCode[Knot[11, Alternating, 215]]
Out[5]=	DTCode[4, 12, 18, 14, 16, 2, 22, 20, 10, 6, 8]
In[6]:=	alex = Alexander[Knot[11, Alternating, 215]][t]
Out[6]=	-4 6 17 27 2 3 4 31 + t - -- + -- - -- - 27 t + 17 t - 6 t + t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 215]][z]
Out[7]=	2 4 6 8 1 + 3 z + z + 2 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 215]}
In[9]:=	{KnotDet[Knot[11, Alternating, 215]], KnotSignature[Knot[11, Alternating, 215]]}
Out[9]=	{133, 4}
In[10]:=	J=Jones[Knot[11, Alternating, 215]][q]
Out[10]=	1 2 3 4 5 6 7 8 9 10 4 - - - 8 q + 14 q - 18 q + 21 q - 21 q + 19 q - 14 q + 8 q - 4 q + q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 215]}
In[12]:=	A2Invariant[Knot[11, Alternating, 215]][q]
Out[12]=	-2 2 4 6 8 10 12 14 18 20 2 - q - 2 q + 2 q + 2 q - 2 q + 5 q - 4 q + 3 q - q + 3 q - 22 24 26 28 30 > 4 q + q - q - q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 215]][a, z]
Out[13]=	2 2 2 2 4 4 4 4 6 6 -2 3 2 z 7 z 10 z 2 z z 7 z 10 z 3 z 2 z 5 z -- + -- + ---- - ---- + ----- - ---- + -- - ---- + ----- - ---- - ---- + ---- - 6 4 8 6 4 2 8 6 4 2 6 4 a a a a a a a a a a a a 6 8 z z > -- + -- 2 4 a a
In[14]:=	Kauffman[Knot[11, Alternating, 215]][a, z]
Out[14]=	2 2 2 2 2 3 2 3 5 z 8 z 4 z z z 2 z 18 z 21 z 6 z 2 z -- + -- - --- - --- - --- - - + --- - ---- - ----- - ----- - ---- - ---- + 6 4 7 5 3 a 10 8 6 4 2 11 a a a a a a a a a a a 3 3 3 3 3 4 4 4 4 4 6 z 23 z 22 z 10 z 3 z z 6 z 11 z 56 z 54 z > ---- + ----- + ----- + ----- + ---- + --- - ---- + ----- + ----- + ----- + 9 7 5 3 a 12 10 8 6 4 a a a a a a a a a 4 5 5 5 5 5 5 6 6 6 16 z 4 z 15 z 26 z 3 z z 3 z 8 z 21 z 64 z > ----- + ---- - ----- - ----- - ---- + -- - ---- + ---- - ----- - ----- - 2 11 9 7 5 3 a 10 8 6 a a a a a a a a a 6 6 7 7 7 7 7 8 8 8 49 z 14 z 12 z 2 z 27 z 12 z z 13 z 19 z 10 z > ----- - ----- + ----- - ---- - ----- - ----- + -- + ----- + ----- + ----- + 4 2 9 7 5 3 a 8 6 4 a a a a a a a a a 8 9 9 9 10 10 4 z 8 z 13 z 5 z 2 z 2 z > ---- + ---- + ----- + ---- + ----- + ----- 2 7 5 3 6 4 a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 215]], Vassiliev[3][Knot[11, Alternating, 215]]}
Out[15]=	{3, 5}
In[16]:=	Kh[Knot[11, Alternating, 215]][q, t]
Out[16]=	3 3 5 1 3 q 5 q 3 q 5 7 7 2 9 q + 6 q + ----- + ---- + -- + --- + ---- + 10 q t + 8 q t + 11 q t + 3 3 2 2 t t q t q t t 9 2 9 3 11 3 11 4 13 4 13 5 > 10 q t + 10 q t + 11 q t + 9 q t + 10 q t + 5 q t + 15 5 15 6 17 6 17 7 19 7 21 8 > 9 q t + 3 q t + 5 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a215

K11a214 K11a216