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

K11a330 K11a332

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 12 18 22 16 4 20 10 8 2 14

Alexander Polynomial:

2t-3 - 11t-2 + 27t-1 - 35 + 27t - 11t2 + 2t3

Conway Polynomial:

1 + z2 + z4 + 2z6

Other knots with the same Alexander/Conway Polynomial:

{10121, K11a41, K11a183, K11a198, ...}

Determinant and Signature:

{115, -2}

Jones Polynomial:

- q-8 + 3q-7 - 7q-6 + 12q-5 - 16q-4 + 19q-3 - 18q-2 + 16q-1 - 12 + 7q - 3q2 + q3

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

{K11a3, K11a51, ...}

A2 (sl(3)) Invariant:

- q-24 + q-22 - 2q-20 - 2q-18 + 4q-16 - 2q-14 + 3q-12 + 2q-10 - q-8 + 3q-6 - 4q-4 + 2q-2 - 1 - 2q2 + 3q4 - q6 + q10

HOMFLY-PT Polynomial:

a-2 + a-2z2 - 1 - 3z2 - 2z4 - a2 - a2z2 + a2z4 + a2z6 + 4a4 + 6a4z2 + 3a4z4 + a4z6 - 2a6 - 2a6z2 - a6z4

Kauffman Polynomial:

- a-2 + 3a-2z2 - 3a-2z4 + a-2z6 - a-1z + 6a-1z3 - 8a-1z5 + 3a-1z7 - 1 + 6z2 - 4z4 - 6z6 + 4z8 - 2az3 - az5 - 5az7 + 4az9 + a2 - 5a2z2 + 7a2z4 - 11a2z6 + 3a2z8 + 2a2z10 + a3z - 14a3z3 + 28a3z5 - 25a3z7 + 10a3z9 + 4a4 - 20a4z2 + 39a4z4 - 28a4z6 + 7a4z8 + 2a4z10 - 3a5z + 6a5z3 + 7a5z5 - 11a5z7 + 6a5z9 + 2a6 - 12a6z2 + 26a6z4 - 21a6z6 + 8a6z8 - 3a7z + 10a7z3 - 13a7z5 + 6a7z7 - 5a8z4 + 3a8z6 - 2a9z3 + a9z5

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

{1, -4}

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 11331. 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                   5 1   j = 1                 7 2     j = -1               9 5       j = -3             10 8         j = -5           9 8           j = -7         7 10             j = -9       5 9               j = -11     2 7                 j = -13   1 5                   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, 331]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 331]]
Out[3]=	PD[X[6, 2, 7, 1], X[12, 4, 13, 3], X[18, 5, 19, 6], X[22, 8, 1, 7], > X[16, 9, 17, 10], X[4, 12, 5, 11], X[20, 13, 21, 14], X[10, 15, 11, 16], > X[8, 17, 9, 18], X[2, 19, 3, 20], X[14, 21, 15, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 331]]
Out[4]=	GaussCode[1, -10, 2, -6, 3, -1, 4, -9, 5, -8, 6, -2, 7, -11, 8, -5, 9, -3, 10, > -7, 11, -4]
In[5]:=	DTCode[Knot[11, Alternating, 331]]
Out[5]=	DTCode[6, 12, 18, 22, 16, 4, 20, 10, 8, 2, 14]
In[6]:=	alex = Alexander[Knot[11, Alternating, 331]][t]
Out[6]=	2 11 27 2 3 -35 + -- - -- + -- + 27 t - 11 t + 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 331]][z]
Out[7]=	2 4 6 1 + z + z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 121], Knot[11, Alternating, 41], Knot[11, Alternating, 183], > Knot[11, Alternating, 198], Knot[11, Alternating, 331]}
In[9]:=	{KnotDet[Knot[11, Alternating, 331]], KnotSignature[Knot[11, Alternating, 331]]}
Out[9]=	{115, -2}
In[10]:=	J=Jones[Knot[11, Alternating, 331]][q]
Out[10]=	-8 3 7 12 16 19 18 16 2 3 -12 - q + -- - -- + -- - -- + -- - -- + -- + 7 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, 3], Knot[11, Alternating, 51], > Knot[11, Alternating, 331]}
In[12]:=	A2Invariant[Knot[11, Alternating, 331]][q]
Out[12]=	-24 -22 2 2 4 2 3 2 -8 3 4 2 -1 - q + q - --- - --- + --- - --- + --- + --- - q + -- - -- + -- - 20 18 16 14 12 10 6 4 2 q q q q q q q q q 2 4 6 10 > 2 q + 3 q - q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 331]][a, z]
Out[13]=	2 -2 2 4 6 2 z 2 2 4 2 6 2 4 -1 + a - a + 4 a - 2 a - 3 z + -- - a z + 6 a z - 2 a z - 2 z + 2 a 2 4 4 4 6 4 2 6 4 6 > a z + 3 a z - a z + a z + a z
In[14]:=	Kauffman[Knot[11, Alternating, 331]][a, z]
Out[14]=	2 -2 2 4 6 z 3 5 7 2 3 z -1 - a + a + 4 a + 2 a - - + a z - 3 a z - 3 a z + 6 z + ---- - a 2 a 3 2 2 4 2 6 2 6 z 3 3 3 5 3 > 5 a z - 20 a z - 12 a z + ---- - 2 a z - 14 a z + 6 a z + a 4 7 3 9 3 4 3 z 2 4 4 4 6 4 > 10 a z - 2 a z - 4 z - ---- + 7 a z + 39 a z + 26 a z - 2 a 5 6 8 4 8 z 5 3 5 5 5 7 5 9 5 6 z > 5 a z - ---- - a z + 28 a z + 7 a z - 13 a z + a z - 6 z + -- - a 2 a 7 2 6 4 6 6 6 8 6 3 z 7 3 7 > 11 a z - 28 a z - 21 a z + 3 a z + ---- - 5 a z - 25 a z - a 5 7 7 7 8 2 8 4 8 6 8 9 > 11 a z + 6 a z + 4 z + 3 a z + 7 a z + 8 a z + 4 a z + 3 9 5 9 2 10 4 10 > 10 a z + 6 a z + 2 a z + 2 a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 331]], Vassiliev[3][Knot[11, Alternating, 331]]}
Out[15]=	{1, -4}
In[16]:=	Kh[Knot[11, Alternating, 331]][q, t]
Out[16]=	8 9 1 2 1 5 2 7 5 9 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 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 7 10 9 8 10 5 t 2 3 2 > ----- + ----- + ----- + ---- + ---- + --- + 7 q t + 2 q t + 5 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 K11a331

K11a330 K11a332