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

K11a51 K11a53

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8394 X14,5,15,6 X12,8,13,7 X2,9,3,10 X18,12,19,11 X22,13,1,14 X20,16,21,15 X10,18,11,17 X16,20,17,19 X6,21,7,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 8 14 12 2 18 22 20 10 16 6

Alexander Polynomial:

- 2t-3 + 13t-2 - 32t-1 + 43 - 32t + 13t2 - 2t3

Conway Polynomial:

1 + 2z2 + z4 - 2z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{137, 0}

Jones Polynomial:

- q-5 + 3q-4 - 7q-3 + 13q-2 - 18q-1 + 22 - 22q + 20q2 - 15q3 + 10q4 - 5q5 + q6

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

{...}

A2 (sl(3)) Invariant:

- q-16 + q-12 - 3q-10 + 3q-8 + 2q-6 - 3q-4 + 4q-2 - 3 + 2q2 + q4 - q6 + 5q8 - 4q10 + q12 + q14 - 3q16 + q18

HOMFLY-PT Polynomial:

- a-4 + a-4z4 + 2a-2 + a-2z2 - a-2z4 - a-2z6 - 1 - z2 - z4 - z6 + 2a2 + 3a2z2 + 2a2z4 - a4 - a4z2

Kauffman Polynomial:

- a-6z4 + a-6z6 + a-5z + 3a-5z3 - 10a-5z5 + 5a-5z7 - a-4 + 11a-4z4 - 21a-4z6 + 9a-4z8 + 7a-3z3 - 9a-3z5 - 8a-3z7 + 7a-3z9 - 2a-2 - 2a-2z2 + 27a-2z4 - 41a-2z6 + 14a-2z8 + 2a-2z10 - 3a-1z + 10a-1z3 - a-1z5 - 18a-1z7 + 12a-1z9 - 1 - z2 + 19z4 - 27z6 + 11z8 + 2z10 - 3az + 10az3 - 9az5 + 5az9 - 2a2 + 4a2z2 - a2z4 - 5a2z6 + 6a2z8 + 2a3z3 - 6a3z5 + 5a3z7 - a4 + 3a4z2 - 5a4z4 + 3a4z6 + a5z - 2a5z3 + a5z5

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

{2, 0}

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 1152. 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 = 13                       1 j = 11                     4   j = 9                   6 1   j = 7                 9 4     j = 5               11 6       j = 3             11 9         j = 1           11 11           j = -1         8 12             j = -3       5 10               j = -5     2 8                 j = -7   1 5                   j = -9   2                     j = -11 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, 52]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 52]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 3, 9, 4], X[14, 5, 15, 6], X[12, 8, 13, 7], > X[2, 9, 3, 10], X[18, 12, 19, 11], X[22, 13, 1, 14], X[20, 16, 21, 15], > X[10, 18, 11, 17], X[16, 20, 17, 19], X[6, 21, 7, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 52]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -11, 4, -2, 5, -9, 6, -4, 7, -3, 8, -10, 9, -6, 10, > -8, 11, -7]
In[5]:=	DTCode[Knot[11, Alternating, 52]]
Out[5]=	DTCode[4, 8, 14, 12, 2, 18, 22, 20, 10, 16, 6]
In[6]:=	alex = Alexander[Knot[11, Alternating, 52]][t]
Out[6]=	2 13 32 2 3 43 - -- + -- - -- - 32 t + 13 t - 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 52]][z]
Out[7]=	2 4 6 1 + 2 z + z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 52]}
In[9]:=	{KnotDet[Knot[11, Alternating, 52]], KnotSignature[Knot[11, Alternating, 52]]}
Out[9]=	{137, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 52]][q]
Out[10]=	-5 3 7 13 18 2 3 4 5 6 22 - q + -- - -- + -- - -- - 22 q + 20 q - 15 q + 10 q - 5 q + q 4 3 2 q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 52]}
In[12]:=	A2Invariant[Knot[11, Alternating, 52]][q]
Out[12]=	-16 -12 3 3 2 3 4 2 4 6 8 10 -3 - q + q - --- + -- + -- - -- + -- + 2 q + q - q + 5 q - 4 q + 10 8 6 4 2 q q q q q 12 14 16 18 > q + q - 3 q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 52]][a, z]
Out[13]=	2 4 4 -4 2 2 4 2 z 2 2 4 2 4 z z -1 - a + -- + 2 a - a - z + -- + 3 a z - a z - z + -- - -- + 2 2 4 2 a a a a 6 2 4 6 z > 2 a z - z - -- 2 a
In[14]:=	Kauffman[Knot[11, Alternating, 52]][a, z]
Out[14]=	2 -4 2 2 4 z 3 z 5 2 2 z 2 2 -1 - a - -- - 2 a - a + -- - --- - 3 a z + a z - z - ---- + 4 a z + 2 5 a 2 a a a 3 3 3 4 4 2 3 z 7 z 10 z 3 3 3 5 3 4 z > 3 a z + ---- + ---- + ----- + 10 a z + 2 a z - 2 a z + 19 z - -- + 5 3 a 6 a a a 4 4 5 5 5 11 z 27 z 2 4 4 4 10 z 9 z z 5 3 5 > ----- + ----- - a z - 5 a z - ----- - ---- - -- - 9 a z - 6 a z + 4 2 5 3 a a a a a 6 6 6 7 7 5 5 6 z 21 z 41 z 2 6 4 6 5 z 8 z > a z - 27 z + -- - ----- - ----- - 5 a z + 3 a z + ---- - ---- - 6 4 2 5 3 a a a a a 7 8 8 9 9 18 z 3 7 8 9 z 14 z 2 8 7 z 12 z 9 > ----- + 5 a z + 11 z + ---- + ----- + 6 a z + ---- + ----- + 5 a z + a 4 2 3 a a a a 10 10 2 z > 2 z + ----- 2 a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 52]], Vassiliev[3][Knot[11, Alternating, 52]]}
Out[15]=	{2, 0}
In[16]:=	Kh[Knot[11, Alternating, 52]][q, t]
Out[16]=	12 1 2 1 5 2 8 5 10 -- + 11 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ---- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t q t 8 3 3 2 5 2 5 3 7 3 7 4 > --- + 11 q t + 11 q t + 9 q t + 11 q t + 6 q t + 9 q t + 4 q t + q t 9 4 9 5 11 5 13 6 > 6 q t + q t + 4 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a52

K11a51 K11a53