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

K11a324 K11a326

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 12 16 22 20 18 2 10 4 14 8

Alexander Polynomial:

- 6t-2 + 24t-1 - 35 + 24t - 6t2

Conway Polynomial:

1 - 6z4

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{95, -2}

Jones Polynomial:

- q-8 + 3q-7 - 6q-6 + 10q-5 - 13q-4 + 15q-3 - 15q-2 + 13q-1 - 9 + 6q - 3q2 + q3

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

{K11a7, ...}

A2 (sl(3)) Invariant:

- q-26 - q-24 + 2q-22 - q-20 + 4q-16 - 2q-14 + q-12 - 2q-8 + q-6 - 3q-4 + 3q-2 + 1 - q2 + 3q4 - q6 - q8 + q10

HOMFLY-PT Polynomial:

a-2z2 + 2 + z2 - z4 - 2a2 - 4a2z2 - 3a2z4 - a4z2 - 2a4z4 + 2a6 + 3a6z2 - a8

Kauffman Polynomial:

a-2z2 - 3a-2z4 + a-2z6 + 4a-1z3 - 9a-1z5 + 3a-1z7 + 2 - 9z2 + 17z4 - 17z6 + 5z8 + 2az - 11az3 + 20az5 - 17az7 + 5az9 + 2a2 - 19a2z2 + 37a2z4 - 21a2z6 + a2z8 + 2a2z10 + 3a3z - 21a3z3 + 45a3z5 - 32a3z7 + 9a3z9 - 6a4z2 + 17a4z4 - 9a4z6 + 2a4z10 + a5z - 6a5z3 + 10a5z5 - 8a5z7 + 4a5z9 - 2a6 + 6a6z2 - 6a6z4 - 3a6z6 + 4a6z8 + a7z - 2a7z3 - 5a7z5 + 4a7z7 - a8 + 3a8z2 - 6a8z4 + 3a8z6 + a9z - 2a9z3 + a9z5

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

{0, -2}

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 11325. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 7                       1 j = 5                     2   j = 3                   4 1   j = 1                 5 2     j = -1               8 4       j = -3             8 6         j = -5           7 7           j = -7         6 8             j = -9       4 7               j = -11     2 6                 j = -13   1 4                   j = -15   2                     j = -17 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, 325]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 325]]
Out[3]=	PD[X[6, 2, 7, 1], X[12, 3, 13, 4], X[16, 5, 17, 6], X[22, 8, 1, 7], > X[20, 10, 21, 9], X[18, 11, 19, 12], X[2, 13, 3, 14], X[10, 15, 11, 16], > X[4, 17, 5, 18], X[14, 19, 15, 20], X[8, 22, 9, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 325]]
Out[4]=	GaussCode[1, -7, 2, -9, 3, -1, 4, -11, 5, -8, 6, -2, 7, -10, 8, -3, 9, -6, 10, > -5, 11, -4]
In[5]:=	DTCode[Knot[11, Alternating, 325]]
Out[5]=	DTCode[6, 12, 16, 22, 20, 18, 2, 10, 4, 14, 8]
In[6]:=	alex = Alexander[Knot[11, Alternating, 325]][t]
Out[6]=	6 24 2 -35 - -- + -- + 24 t - 6 t 2 t t
In[7]:=	Conway[Knot[11, Alternating, 325]][z]
Out[7]=	4 1 - 6 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 325]}
In[9]:=	{KnotDet[Knot[11, Alternating, 325]], KnotSignature[Knot[11, Alternating, 325]]}
Out[9]=	{95, -2}
In[10]:=	J=Jones[Knot[11, Alternating, 325]][q]
Out[10]=	-8 3 6 10 13 15 15 13 2 3 -9 - q + -- - -- + -- - -- + -- - -- + -- + 6 q - 3 q + q 7 6 5 4 3 2 q q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 7], Knot[11, Alternating, 325]}
In[12]:=	A2Invariant[Knot[11, Alternating, 325]][q]
Out[12]=	-26 -24 2 -20 4 2 -12 2 -6 3 3 2 1 - q - q + --- - q + --- - --- + q - -- + q - -- + -- - q + 22 16 14 8 4 2 q q q q q q 4 6 8 10 > 3 q - q - q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 325]][a, z]
Out[13]=	2 2 6 8 2 z 2 2 4 2 6 2 4 2 4 2 - 2 a + 2 a - a + z + -- - 4 a z - a z + 3 a z - z - 3 a z - 2 a 4 4 > 2 a z
In[14]:=	Kauffman[Knot[11, Alternating, 325]][a, z]
Out[14]=	2 2 6 8 3 5 7 9 2 z 2 + 2 a - 2 a - a + 2 a z + 3 a z + a z + a z + a z - 9 z + -- - 2 a 3 2 2 4 2 6 2 8 2 4 z 3 3 3 > 19 a z - 6 a z + 6 a z + 3 a z + ---- - 11 a z - 21 a z - a 4 5 3 7 3 9 3 4 3 z 2 4 4 4 > 6 a z - 2 a z - 2 a z + 17 z - ---- + 37 a z + 17 a z - 2 a 5 6 4 8 4 9 z 5 3 5 5 5 7 5 > 6 a z - 6 a z - ---- + 20 a z + 45 a z + 10 a z - 5 a z + a 6 7 9 5 6 z 2 6 4 6 6 6 8 6 3 z > a z - 17 z + -- - 21 a z - 9 a z - 3 a z + 3 a z + ---- - 2 a a 7 3 7 5 7 7 7 8 2 8 6 8 9 > 17 a z - 32 a z - 8 a z + 4 a z + 5 z + a z + 4 a z + 5 a z + 3 9 5 9 2 10 4 10 > 9 a z + 4 a z + 2 a z + 2 a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 325]], Vassiliev[3][Knot[11, Alternating, 325]]}
Out[15]=	{0, -2}
In[16]:=	Kh[Knot[11, Alternating, 325]][q, t]
Out[16]=	6 8 1 2 1 4 2 6 4 7 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 3 q 17 7 15 6 13 6 13 5 11 5 11 4 9 4 9 3 q q t q t q t q t q t q t q t q t 6 8 7 7 8 4 t 2 3 2 > ----- + ----- + ----- + ---- + ---- + --- + 5 q t + 2 q t + 4 q t + 7 3 7 2 5 2 5 3 q q t q t q t q t q t 3 3 5 3 7 4 > q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a325

K11a324 K11a326