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

K11a329 K11a331

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 X8,15,9,16 X10,17,11,18 X2,19,3,20 X14,21,15,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 12 18 22 16 4 20 8 10 2 14

Alexander Polynomial:

t-4 - 5t-3 + 12t-2 - 17t-1 + 19 - 17t + 12t2 - 5t3 + t4

Conway Polynomial:

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

Other knots with the same Alexander/Conway Polynomial:

{K11a40, ...}

Determinant and Signature:

{89, -4}

Jones Polynomial:

- q-9 + 3q-8 - 6q-7 + 9q-6 - 12q-5 + 14q-4 - 13q-3 + 12q-2 - 9q-1 + 6 - 3q + q2

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

{...}

A2 (sl(3)) Invariant:

- q-26 + q-24 - 2q-22 - q-16 + 4q-14 - q-12 + 3q-10 - q-6 + q-4 - 2q-2 + 1 + q6

HOMFLY-PT Polynomial:

2 + 3z2 + z4 - 5a2 - 12a2z2 - 9a2z4 - 2a2z6 + 7a4 + 16a4z2 + 14a4z4 + 6a4z6 + a4z8 - 3a6 - 5a6z2 - 4a6z4 - a6z6

Kauffman Polynomial:

2 - 7z2 + 9z4 - 5z6 + z8 + az - 13az3 + 24az5 - 15az7 + 3az9 + 5a2 - 27a2z2 + 43a2z4 - 18a2z6 - 3a2z8 + 2a2z10 - 14a3z3 + 49a3z5 - 42a3z7 + 10a3z9 + 7a4 - 35a4z2 + 72a4z4 - 52a4z6 + 7a4z8 + 2a4z10 - 4a5z + 13a5z3 - 3a5z5 - 16a5z7 + 7a5z9 + 3a6 - 12a6z2 + 23a6z4 - 30a6z6 + 11a6z8 - 2a7z + 9a7z3 - 22a7z5 + 11a7z7 + 3a8z2 - 12a8z4 + 9a8z6 + a9z - 4a9z3 + 6a9z5 + 3a10z4 + a11z3

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

{2, -5}

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 11330. 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 = 5                       1 j = 3                     2   j = 1                   4 1   j = -1                 5 2     j = -3               7 4       j = -5             7 6         j = -7           7 6           j = -9         5 7             j = -11       4 7               j = -13     2 5                 j = -15   1 4                   j = -17   2                     j = -19 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, 330]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 330]]
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[8, 15, 9, 16], > X[10, 17, 11, 18], X[2, 19, 3, 20], X[14, 21, 15, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 330]]
Out[4]=	GaussCode[1, -10, 2, -6, 3, -1, 4, -8, 5, -9, 6, -2, 7, -11, 8, -5, 9, -3, 10, > -7, 11, -4]
In[5]:=	DTCode[Knot[11, Alternating, 330]]
Out[5]=	DTCode[6, 12, 18, 22, 16, 4, 20, 8, 10, 2, 14]
In[6]:=	alex = Alexander[Knot[11, Alternating, 330]][t]
Out[6]=	-4 5 12 17 2 3 4 19 + t - -- + -- - -- - 17 t + 12 t - 5 t + t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 330]][z]
Out[7]=	2 4 6 8 1 + 2 z + 2 z + 3 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 40], Knot[11, Alternating, 330]}
In[9]:=	{KnotDet[Knot[11, Alternating, 330]], KnotSignature[Knot[11, Alternating, 330]]}
Out[9]=	{89, -4}
In[10]:=	J=Jones[Knot[11, Alternating, 330]][q]
Out[10]=	-9 3 6 9 12 14 13 12 9 2 6 - q + -- - -- + -- - -- + -- - -- + -- - - - 3 q + q 8 7 6 5 4 3 2 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, 330]}
In[12]:=	A2Invariant[Knot[11, Alternating, 330]][q]
Out[12]=	-26 -24 2 -16 4 -12 3 -6 -4 2 6 1 - q + q - --- - q + --- - q + --- - q + q - -- + q 22 14 10 2 q q q q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 330]][a, z]
Out[13]=	2 4 6 2 2 2 4 2 6 2 4 2 4 2 - 5 a + 7 a - 3 a + 3 z - 12 a z + 16 a z - 5 a z + z - 9 a z + 4 4 6 4 2 6 4 6 6 6 4 8 > 14 a z - 4 a z - 2 a z + 6 a z - a z + a z
In[14]:=	Kauffman[Knot[11, Alternating, 330]][a, z]
Out[14]=	2 4 6 5 7 9 2 2 2 2 + 5 a + 7 a + 3 a + a z - 4 a z - 2 a z + a z - 7 z - 27 a z - 4 2 6 2 8 2 3 3 3 5 3 7 3 > 35 a z - 12 a z + 3 a z - 13 a z - 14 a z + 13 a z + 9 a z - 9 3 11 3 4 2 4 4 4 6 4 8 4 > 4 a z + a z + 9 z + 43 a z + 72 a z + 23 a z - 12 a z + 10 4 5 3 5 5 5 7 5 9 5 6 > 3 a z + 24 a z + 49 a z - 3 a z - 22 a z + 6 a z - 5 z - 2 6 4 6 6 6 8 6 7 3 7 5 7 > 18 a z - 52 a z - 30 a z + 9 a z - 15 a z - 42 a z - 16 a z + 7 7 8 2 8 4 8 6 8 9 3 9 > 11 a z + z - 3 a z + 7 a z + 11 a z + 3 a z + 10 a z + 5 9 2 10 4 10 > 7 a z + 2 a z + 2 a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 330]], Vassiliev[3][Knot[11, Alternating, 330]]}
Out[15]=	{2, -5}
In[16]:=	Kh[Knot[11, Alternating, 330]][q, t]
Out[16]=	6 7 1 2 1 4 2 5 4 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 5 3 19 7 17 6 15 6 15 5 13 5 13 4 11 4 q q q t q t q t q t q t q t q t 2 7 5 7 7 6 7 4 t 5 t 2 t 2 > ------ + ----- + ----- + ----- + ---- + ---- + --- + --- + ---- + 4 q t + 11 3 9 3 9 2 7 2 7 5 3 q q q t q t q t q t q t q t q 3 3 3 5 4 > q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a330

K11a329 K11a331