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

K11a325 K11a327

Knotscape

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DrawMorseLink

PD Presentation:

X6271 X12,4,13,3 X18,5,19,6 X14,8,15,7 X22,10,1,9 X4,12,5,11 X20,13,21,14 X8,16,9,15 X10,17,11,18 X2,19,3,20 X16,21,17,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 12 18 14 22 4 20 8 10 2 16

Alexander Polynomial:

t-4 - 6t-3 + 19t-2 - 36t-1 + 45 - 36t + 19t2 - 6t3 + t4

Conway Polynomial:

1 + 2z2 + 3z4 + 2z6 + z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{169, 0}

Jones Polynomial:

- q-5 + 4q-4 - 10q-3 + 17q-2 - 23q-1 + 28 - 27q + 24q2 - 18q3 + 11q4 - 5q5 + q6

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

{...}

A2 (sl(3)) Invariant:

- q-14 + 2q-12 - 4q-10 + 2q-8 + q-6 - 4q-4 + 7q-2 - 3 + 5q2 - 2q6 + 4q8 - 5q10 + 2q12 - 2q16 + q18

HOMFLY-PT Polynomial:

a-4z2 + a-4z4 - 2a-2 - 6a-2z2 - 6a-2z4 - 2a-2z6 + 5 + 11z2 + 11z4 + 5z6 + z8 - 2a2 - 4a2z2 - 3a2z4 - a2z6

Kauffman Polynomial:

- a-6z4 + a-6z6 + a-5z + 3a-5z3 - 9a-5z5 + 5a-5z7 - 2a-4z2 + 11a-4z4 - 21a-4z6 + 10a-4z8 + a-3z + 2a-3z5 - 16a-3z7 + 10a-3z9 + 2a-2 - 14a-2z2 + 39a-2z4 - 47a-2z6 + 13a-2z8 + 4a-2z10 - a-1z - 5a-1z3 + 28a-1z5 - 44a-1z7 + 21a-1z9 + 5 - 22z2 + 53z4 - 53z6 + 16z8 + 4z10 - 3az + 6az3 + 3az5 - 14az7 + 11az9 + 2a2 - 10a2z2 + 22a2z4 - 24a2z6 + 13a2z8 - 2a3z + 7a3z3 - 13a3z5 + 9a3z7 - 4a4z4 + 4a4z6 - a5z3 + 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 11326. 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                   7 1   j = 7                 11 4     j = 5               13 7       j = 3             14 11         j = 1           14 13           j = -1         10 15             j = -3       7 13               j = -5     3 10                 j = -7   1 7                   j = -9   3                     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, 326]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 326]]
Out[3]=	PD[X[6, 2, 7, 1], X[12, 4, 13, 3], X[18, 5, 19, 6], X[14, 8, 15, 7], > X[22, 10, 1, 9], X[4, 12, 5, 11], X[20, 13, 21, 14], X[8, 16, 9, 15], > X[10, 17, 11, 18], X[2, 19, 3, 20], X[16, 21, 17, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 326]]
Out[4]=	GaussCode[1, -10, 2, -6, 3, -1, 4, -8, 5, -9, 6, -2, 7, -4, 8, -11, 9, -3, 10, > -7, 11, -5]
In[5]:=	DTCode[Knot[11, Alternating, 326]]
Out[5]=	DTCode[6, 12, 18, 14, 22, 4, 20, 8, 10, 2, 16]
In[6]:=	alex = Alexander[Knot[11, Alternating, 326]][t]
Out[6]=	-4 6 19 36 2 3 4 45 + t - -- + -- - -- - 36 t + 19 t - 6 t + t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 326]][z]
Out[7]=	2 4 6 8 1 + 2 z + 3 z + 2 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 326]}
In[9]:=	{KnotDet[Knot[11, Alternating, 326]], KnotSignature[Knot[11, Alternating, 326]]}
Out[9]=	{169, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 326]][q]
Out[10]=	-5 4 10 17 23 2 3 4 5 6 28 - q + -- - -- + -- - -- - 27 q + 24 q - 18 q + 11 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, 326]}
In[12]:=	A2Invariant[Knot[11, Alternating, 326]][q]
Out[12]=	-14 2 4 2 -6 4 7 2 6 8 10 -3 - q + --- - --- + -- + q - -- + -- + 5 q - 2 q + 4 q - 5 q + 12 10 8 4 2 q q q q q 12 16 18 > 2 q - 2 q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 326]][a, z]
Out[13]=	2 2 4 4 2 2 2 z 6 z 2 2 4 z 6 z 2 4 5 - -- - 2 a + 11 z + -- - ---- - 4 a z + 11 z + -- - ---- - 3 a z + 2 4 2 4 2 a a a a a 6 6 2 z 2 6 8 > 5 z - ---- - a z + z 2 a
In[14]:=	Kauffman[Knot[11, Alternating, 326]][a, z]
Out[14]=	2 2 2 2 z z z 3 2 2 z 14 z 5 + -- + 2 a + -- + -- - - - 3 a z - 2 a z - 22 z - ---- - ----- - 2 5 3 a 4 2 a a a a a 3 3 4 4 2 2 3 z 5 z 3 3 3 5 3 4 z 11 z > 10 a z + ---- - ---- + 6 a z + 7 a z - a z + 53 z - -- + ----- + 5 a 6 4 a a a 4 5 5 5 39 z 2 4 4 4 9 z 2 z 28 z 5 3 5 > ----- + 22 a z - 4 a z - ---- + ---- + ----- + 3 a z - 13 a z + 2 5 3 a a a a 6 6 6 7 7 5 5 6 z 21 z 47 z 2 6 4 6 5 z 16 z > a z - 53 z + -- - ----- - ----- - 24 a z + 4 a z + ---- - ----- - 6 4 2 5 3 a a a a a 7 8 8 9 44 z 7 3 7 8 10 z 13 z 2 8 10 z > ----- - 14 a z + 9 a z + 16 z + ----- + ----- + 13 a z + ----- + a 4 2 3 a a a 9 10 21 z 9 10 4 z > ----- + 11 a z + 4 z + ----- a 2 a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 326]], Vassiliev[3][Knot[11, Alternating, 326]]}
Out[15]=	{2, 0}
In[16]:=	Kh[Knot[11, Alternating, 326]][q, t]
Out[16]=	15 1 3 1 7 3 10 7 13 -- + 14 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 10 3 3 2 5 2 5 3 7 3 > --- + 13 q t + 14 q t + 11 q t + 13 q t + 7 q t + 11 q t + q t 7 4 9 4 9 5 11 5 13 6 > 4 q t + 7 q t + q t + 4 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a326

K11a325 K11a327