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

K11a295 K11a297

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 10 16 22 18 2 20 8 4 14 12

Alexander Polynomial:

- 6t-2 + 28t-1 - 43 + 28t - 6t2

Conway Polynomial:

1 + 4z2 - 6z4

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{111, -2}

Jones Polynomial:

- q-10 + 3q-9 - 7q-8 + 11q-7 - 15q-6 + 18q-5 - 17q-4 + 16q-3 - 12q-2 + 7q-1 - 3 + q

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

{...}

A2 (sl(3)) Invariant:

- q-32 - q-30 + 2q-28 - 2q-26 - q-24 + 4q-22 - 3q-20 + q-18 + q-16 - q-14 + 3q-12 - 2q-10 + 3q-8 - 3q-4 + 4q-2 - 1 - q2 + q4

HOMFLY-PT Polynomial:

z2 + a2 + 2a2z2 - a2z4 - 2a4z2 - 3a4z4 - 2a6z4 + a8 + 3a8z2 - a10

Kauffman Polynomial:

- z2 + z4 - 2az3 + 3az5 - a2 + 4a2z2 - 6a2z4 + 6a2z6 - a3z + 11a3z3 - 14a3z5 + 9a3z7 + 7a4z4 - 14a4z6 + 9a4z8 - a5z + 7a5z3 - 8a5z5 - 5a5z7 + 6a5z9 - 10a6z2 + 25a6z4 - 30a6z6 + 8a6z8 + 2a6z10 + 3a7z - 11a7z3 + 21a7z5 - 27a7z7 + 10a7z9 + a8 - 9a8z2 + 23a8z4 - 21a8z6 + 2a8z8 + 2a8z10 + a9z + 8a9z5 - 12a9z7 + 4a9z9 + a10 - 4a10z2 + 12a10z4 - 11a10z6 + 3a10z8 - 2a11z + 5a11z3 - 4a11z5 + a11z7

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

{4, -10}

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 11296. 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 = 3                       1 j = 1                     2   j = -1                   5 1   j = -3                 8 3     j = -5               8 4       j = -7             9 8         j = -9           9 8           j = -11         6 9             j = -13       5 9               j = -15     2 6                 j = -17   1 5                   j = -19   2                     j = -21 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, 296]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 296]]
Out[3]=	PD[X[6, 2, 7, 1], X[10, 3, 11, 4], X[16, 5, 17, 6], X[22, 8, 1, 7], > X[18, 9, 19, 10], X[2, 11, 3, 12], X[20, 13, 21, 14], X[8, 15, 9, 16], > X[4, 17, 5, 18], X[14, 19, 15, 20], X[12, 21, 13, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 296]]
Out[4]=	GaussCode[1, -6, 2, -9, 3, -1, 4, -8, 5, -2, 6, -11, 7, -10, 8, -3, 9, -5, 10, > -7, 11, -4]
In[5]:=	DTCode[Knot[11, Alternating, 296]]
Out[5]=	DTCode[6, 10, 16, 22, 18, 2, 20, 8, 4, 14, 12]
In[6]:=	alex = Alexander[Knot[11, Alternating, 296]][t]
Out[6]=	6 28 2 -43 - -- + -- + 28 t - 6 t 2 t t
In[7]:=	Conway[Knot[11, Alternating, 296]][z]
Out[7]=	2 4 1 + 4 z - 6 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 296]}
In[9]:=	{KnotDet[Knot[11, Alternating, 296]], KnotSignature[Knot[11, Alternating, 296]]}
Out[9]=	{111, -2}
In[10]:=	J=Jones[Knot[11, Alternating, 296]][q]
Out[10]=	-10 3 7 11 15 18 17 16 12 7 -3 - q + -- - -- + -- - -- + -- - -- + -- - -- + - + q 9 8 7 6 5 4 3 2 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, 296]}
In[12]:=	A2Invariant[Knot[11, Alternating, 296]][q]
Out[12]=	-32 -30 2 2 -24 4 3 -18 -16 -14 3 -1 - q - q + --- - --- - q + --- - --- + q + q - q + --- - 28 26 22 20 12 q q q q q 2 3 3 4 2 4 > --- + -- - -- + -- - q + q 10 8 4 2 q q q q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 296]][a, z]
Out[13]=	2 8 10 2 2 2 4 2 8 2 2 4 4 4 6 4 a + a - a + z + 2 a z - 2 a z + 3 a z - a z - 3 a z - 2 a z
In[14]:=	Kauffman[Knot[11, Alternating, 296]][a, z]
Out[14]=	2 8 10 3 5 7 9 11 2 2 2 -a + a + a - a z - a z + 3 a z + a z - 2 a z - z + 4 a z - 6 2 8 2 10 2 3 3 3 5 3 7 3 > 10 a z - 9 a z - 4 a z - 2 a z + 11 a z + 7 a z - 11 a z + 11 3 4 2 4 4 4 6 4 8 4 10 4 > 5 a z + z - 6 a z + 7 a z + 25 a z + 23 a z + 12 a z + 5 3 5 5 5 7 5 9 5 11 5 2 6 > 3 a z - 14 a z - 8 a z + 21 a z + 8 a z - 4 a z + 6 a z - 4 6 6 6 8 6 10 6 3 7 5 7 7 7 > 14 a z - 30 a z - 21 a z - 11 a z + 9 a z - 5 a z - 27 a z - 9 7 11 7 4 8 6 8 8 8 10 8 5 9 > 12 a z + a z + 9 a z + 8 a z + 2 a z + 3 a z + 6 a z + 7 9 9 9 6 10 8 10 > 10 a z + 4 a z + 2 a z + 2 a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 296]], Vassiliev[3][Knot[11, Alternating, 296]]}
Out[15]=	{4, -10}
In[16]:=	Kh[Knot[11, Alternating, 296]][q, t]
Out[16]=	3 5 1 2 1 5 2 6 5 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 3 q 21 9 19 8 17 8 17 7 15 7 15 6 13 6 q q t q t q t q t q t q t q t 9 6 9 9 8 9 8 8 4 > ------ + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ---- + 13 5 11 5 11 4 9 4 9 3 7 3 7 2 5 2 5 q t q t q t q t q t q t q t q t q t 8 t 3 2 > ---- + - + 2 q t + q t 3 q q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a296

K11a295 K11a297