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

K11a320 K11a322

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 12 16 22 18 20 2 8 4 10 14

Alexander Polynomial:

- 3t-3 + 15t-2 - 27t-1 + 31 - 27t + 15t2 - 3t3

Conway Polynomial:

1 + 6z2 - 3z4 - 3z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{121, -4}

Jones Polynomial:

- q-11 + 3q-10 - 8q-9 + 13q-8 - 17q-7 + 20q-6 - 19q-5 + 17q-4 - 12q-3 + 7q-2 - 3q-1 + 1

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

{...}

A2 (sl(3)) Invariant:

- q-34 - q-32 + q-30 - 3q-28 + 2q-26 + q-24 - 2q-22 + 4q-20 - 3q-18 + 3q-16 - q-12 + 4q-10 - 3q-8 + 2q-6 - q-2 + 1

HOMFLY-PT Polynomial:

2a2z2 + a2z4 + 2a4 + 3a4z2 - a4z4 - a4z6 - 2a6 - 5a6z2 - 6a6z4 - 2a6z6 + 3a8 + 7a8z2 + 3a8z4 - 2a10 - a10z2

Kauffman Polynomial:

2a2z2 - 3a2z4 + a2z6 + 5a3z3 - 8a3z5 + 3a3z7 + 2a4 - 7a4z2 + 12a4z4 - 13a4z6 + 5a4z8 + a5z - 8a5z3 + 11a5z5 - 11a5z7 + 5a5z9 + 2a6 - 11a6z2 + 20a6z4 - 19a6z6 + 5a6z8 + 2a6z10 + 2a7z - 11a7z3 + 20a7z5 - 21a7z7 + 10a7z9 + 3a8 - 8a8z2 + 16a8z4 - 18a8z6 + 7a8z8 + 2a8z10 - 4a9z + 12a9z3 - 10a9z5 - a9z7 + 5a9z9 + 2a10 - 5a10z2 + 7a10z4 - 10a10z6 + 7a10z8 - 4a11z + 8a11z3 - 10a11z5 + 6a11z7 + a12z2 - 4a12z4 + 3a12z6 + a13z - 2a13z3 + a13z5

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

{6, -16}

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 11321. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 1                       1 j = -1                     2   j = -3                   5 1   j = -5                 8 3     j = -7               9 4       j = -9             10 8         j = -11           10 9           j = -13         7 10             j = -15       6 10               j = -17     2 7                 j = -19   1 6                   j = -21   2                     j = -23 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, 321]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 321]]
Out[3]=	PD[X[6, 2, 7, 1], X[12, 3, 13, 4], X[16, 5, 17, 6], X[22, 8, 1, 7], > X[18, 9, 19, 10], X[20, 11, 21, 12], X[2, 13, 3, 14], X[8, 15, 9, 16], > X[4, 17, 5, 18], X[10, 19, 11, 20], X[14, 21, 15, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 321]]
Out[4]=	GaussCode[1, -7, 2, -9, 3, -1, 4, -8, 5, -10, 6, -2, 7, -11, 8, -3, 9, -5, 10, > -6, 11, -4]
In[5]:=	DTCode[Knot[11, Alternating, 321]]
Out[5]=	DTCode[6, 12, 16, 22, 18, 20, 2, 8, 4, 10, 14]
In[6]:=	alex = Alexander[Knot[11, Alternating, 321]][t]
Out[6]=	3 15 27 2 3 31 - -- + -- - -- - 27 t + 15 t - 3 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 321]][z]
Out[7]=	2 4 6 1 + 6 z - 3 z - 3 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 321]}
In[9]:=	{KnotDet[Knot[11, Alternating, 321]], KnotSignature[Knot[11, Alternating, 321]]}
Out[9]=	{121, -4}
In[10]:=	J=Jones[Knot[11, Alternating, 321]][q]
Out[10]=	-11 3 8 13 17 20 19 17 12 7 3 1 - q + --- - -- + -- - -- + -- - -- + -- - -- + -- - - 10 9 8 7 6 5 4 3 2 q q q 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, 321]}
In[12]:=	A2Invariant[Knot[11, Alternating, 321]][q]
Out[12]=	-34 -32 -30 3 2 -24 2 4 3 3 -12 1 - q - q + q - --- + --- + q - --- + --- - --- + --- - q + 28 26 22 20 18 16 q q q q q q 4 3 2 -2 > --- - -- + -- - q 10 8 6 q q q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 321]][a, z]
Out[13]=	4 6 8 10 2 2 4 2 6 2 8 2 10 2 2 a - 2 a + 3 a - 2 a + 2 a z + 3 a z - 5 a z + 7 a z - a z + 2 4 4 4 6 4 8 4 4 6 6 6 > a z - a z - 6 a z + 3 a z - a z - 2 a z
In[14]:=	Kauffman[Knot[11, Alternating, 321]][a, z]
Out[14]=	4 6 8 10 5 7 9 11 13 2 a + 2 a + 3 a + 2 a + a z + 2 a z - 4 a z - 4 a z + a z + 2 2 4 2 6 2 8 2 10 2 12 2 3 3 > 2 a z - 7 a z - 11 a z - 8 a z - 5 a z + a z + 5 a z - 5 3 7 3 9 3 11 3 13 3 2 4 4 4 > 8 a z - 11 a z + 12 a z + 8 a z - 2 a z - 3 a z + 12 a z + 6 4 8 4 10 4 12 4 3 5 5 5 7 5 > 20 a z + 16 a z + 7 a z - 4 a z - 8 a z + 11 a z + 20 a z - 9 5 11 5 13 5 2 6 4 6 6 6 8 6 > 10 a z - 10 a z + a z + a z - 13 a z - 19 a z - 18 a z - 10 6 12 6 3 7 5 7 7 7 9 7 11 7 > 10 a z + 3 a z + 3 a z - 11 a z - 21 a z - a z + 6 a z + 4 8 6 8 8 8 10 8 5 9 7 9 9 9 > 5 a z + 5 a z + 7 a z + 7 a z + 5 a z + 10 a z + 5 a z + 6 10 8 10 > 2 a z + 2 a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 321]], Vassiliev[3][Knot[11, Alternating, 321]]}
Out[15]=	{6, -16}
In[16]:=	Kh[Knot[11, Alternating, 321]][q, t]
Out[16]=	3 5 1 2 1 6 2 7 6 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 5 3 23 9 21 8 19 8 19 7 17 7 17 6 15 6 q q q t q t q t q t q t q t q t 10 7 10 10 9 10 8 9 4 > ------ + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- + 15 5 13 5 13 4 11 4 11 3 9 3 9 2 7 2 7 q t q t q t q t q t q t q t q t q t 8 t 2 t 2 > ---- + -- + --- + q t 5 3 q q t q

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a321

K11a320 K11a322