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

K11a251 K11a253

Knotscape

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DrawMorseLink

PD Presentation:

X6271 X8493 X12,5,13,6 X2837 X18,9,19,10 X16,12,17,11 X4,13,5,14 X20,16,21,15 X22,17,1,18 X14,20,15,19 X10,21,11,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 8 12 2 18 16 4 20 22 14 10

Alexander Polynomial:

- t-4 + 6t-3 - 16t-2 + 27t-1 - 31 + 27t - 16t2 + 6t3 - t4

Conway Polynomial:

1 + z2 - 2z6 - z8

Other knots with the same Alexander/Conway Polynomial:

{K11a131, K11a254, ...}

Determinant and Signature:

{131, 2}

Jones Polynomial:

- q-4 + 3q-3 - 7q-2 + 13q-1 - 17 + 21q - 21q2 + 19q3 - 15q4 + 9q5 - 4q6 + q7

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

{K11a254, ...}

A2 (sl(3)) Invariant:

- q-12 - 2q-6 + 4q-4 - q-2 + 3 + 3q2 - 3q4 + 4q6 - 5q8 + 2q10 - q12 - 2q14 + 3q16 - 2q18 + q20

HOMFLY-PT Polynomial:

a-4 + 3a-4z2 + 3a-4z4 + a-4z6 - 4a-2 - 10a-2z2 - 10a-2z4 - 5a-2z6 - a-2z8 + 6 + 11z2 + 8z4 + 2z6 - 2a2 - 3a2z2 - a2z4

Kauffman Polynomial:

a-8z4 - a-7z3 + 4a-7z5 + 2a-6z2 - 7a-6z4 + 9a-6z6 - 2a-5z + 11a-5z3 - 20a-5z5 + 14a-5z7 + a-4 - 3a-4z2 + 10a-4z4 - 22a-4z6 + 14a-4z8 - 6a-3z + 23a-3z3 - 27a-3z5 - a-3z7 + 8a-3z9 + 4a-2 - 19a-2z2 + 47a-2z4 - 55a-2z6 + 17a-2z8 + 2a-2z10 - 7a-1z + 15a-1z3 + 3a-1z5 - 26a-1z7 + 12a-1z9 + 6 - 22z2 + 43z4 - 35z6 + 6z8 + 2z10 - 5az + 9az3 + 2az5 - 10az7 + 4az9 + 2a2 - 8a2z2 + 14a2z4 - 11a2z6 + 3a2z8 - 2a3z + 5a3z3 - 4a3z5 + a3z7

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

{1, -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=2 is the signature of 11252. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 15                       1 j = 13                     3   j = 11                   6 1   j = 9                 9 3     j = 7               10 6       j = 5             11 9         j = 3           10 10           j = 1         8 12             j = -1       5 9               j = -3     2 8                 j = -5   1 5                   j = -7   2                     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, 252]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 252]]
Out[3]=	PD[X[6, 2, 7, 1], X[8, 4, 9, 3], X[12, 5, 13, 6], X[2, 8, 3, 7], > X[18, 9, 19, 10], X[16, 12, 17, 11], X[4, 13, 5, 14], X[20, 16, 21, 15], > X[22, 17, 1, 18], X[14, 20, 15, 19], X[10, 21, 11, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 252]]
Out[4]=	GaussCode[1, -4, 2, -7, 3, -1, 4, -2, 5, -11, 6, -3, 7, -10, 8, -6, 9, -5, 10, > -8, 11, -9]
In[5]:=	DTCode[Knot[11, Alternating, 252]]
Out[5]=	DTCode[6, 8, 12, 2, 18, 16, 4, 20, 22, 14, 10]
In[6]:=	alex = Alexander[Knot[11, Alternating, 252]][t]
Out[6]=	-4 6 16 27 2 3 4 -31 - t + -- - -- + -- + 27 t - 16 t + 6 t - t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 252]][z]
Out[7]=	2 6 8 1 + z - 2 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 131], Knot[11, Alternating, 252], > Knot[11, Alternating, 254]}
In[9]:=	{KnotDet[Knot[11, Alternating, 252]], KnotSignature[Knot[11, Alternating, 252]]}
Out[9]=	{131, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 252]][q]
Out[10]=	-4 3 7 13 2 3 4 5 6 7 -17 - q + -- - -- + -- + 21 q - 21 q + 19 q - 15 q + 9 q - 4 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, 252], Knot[11, Alternating, 254]}
In[12]:=	A2Invariant[Knot[11, Alternating, 252]][q]
Out[12]=	-12 2 4 -2 2 4 6 8 10 12 14 3 - q - -- + -- - q + 3 q - 3 q + 4 q - 5 q + 2 q - q - 2 q + 6 4 q q 16 18 20 > 3 q - 2 q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 252]][a, z]
Out[13]=	2 2 4 4 -4 4 2 2 3 z 10 z 2 2 4 3 z 10 z 6 + a - -- - 2 a + 11 z + ---- - ----- - 3 a z + 8 z + ---- - ----- - 2 4 2 4 2 a a a a a 6 6 8 2 4 6 z 5 z z > a z + 2 z + -- - ---- - -- 4 2 2 a a a
In[14]:=	Kauffman[Knot[11, Alternating, 252]][a, z]
Out[14]=	2 2 -4 4 2 2 z 6 z 7 z 3 2 2 z 3 z 6 + a + -- + 2 a - --- - --- - --- - 5 a z - 2 a z - 22 z + ---- - ---- - 2 5 3 a 6 4 a a a a a 2 3 3 3 3 19 z 2 2 z 11 z 23 z 15 z 3 3 3 4 > ----- - 8 a z - -- + ----- + ----- + ----- + 9 a z + 5 a z + 43 z + 2 7 5 3 a a a a a 4 4 4 4 5 5 5 5 z 7 z 10 z 47 z 2 4 4 z 20 z 27 z 3 z > -- - ---- + ----- + ----- + 14 a z + ---- - ----- - ----- + ---- + 8 6 4 2 7 5 3 a a a a a a a a 6 6 6 7 7 5 3 5 6 9 z 22 z 55 z 2 6 14 z z > 2 a z - 4 a z - 35 z + ---- - ----- - ----- - 11 a z + ----- - -- - 6 4 2 5 3 a a a a a 7 8 8 9 9 26 z 7 3 7 8 14 z 17 z 2 8 8 z 12 z > ----- - 10 a z + a z + 6 z + ----- + ----- + 3 a z + ---- + ----- + a 4 2 3 a a a a 10 9 10 2 z > 4 a z + 2 z + ----- 2 a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 252]], Vassiliev[3][Knot[11, Alternating, 252]]}
Out[15]=	{1, -1}
In[16]:=	Kh[Knot[11, Alternating, 252]][q, t]
Out[16]=	3 1 2 1 5 2 8 5 9 12 q + 10 q + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + 9 5 7 4 5 4 5 3 3 3 3 2 2 q t q t q t q t q t q t q t q t 8 q 3 5 5 2 7 2 7 3 9 3 > --- + 10 q t + 11 q t + 9 q t + 10 q t + 6 q t + 9 q t + t 9 4 11 4 11 5 13 5 15 6 > 3 q t + 6 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a252

K11a251 K11a253