10 133

10_132

10_134

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10 133 Further Notes and Views

Knot presentations

Planar diagram presentation	X1425 X3849 X14,9,15,10 X5,13,6,12 X13,7,14,6 X18,11,19,12 X20,15,1,16 X16,19,17,20 X10,17,11,18 X7283
Gauss code	-1, 10, -2, 1, -4, 5, -10, 2, 3, -9, 6, 4, -5, -3, 7, -8, 9, -6, 8, -7
Dowker-Thistlethwaite code	4 8 12 2 -14 -18 6 -20 -10 -16
Conway Notation	[23,21,2-]

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 2 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-11][1] Hyperbolic Volume 7.7983 A-Polynomial See Data:10 133/A-polynomial

[edit Notes for 10 133's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $2$ Rasmussen s-Invariant -2

[edit Notes for 10 133's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{2}+5t-7+5t^{-1}-t^{-2}$
Conway polynomial	$-z^{4}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 19, -2 }
Jones polynomial	$q^{-1}-q^{-2}+3q^{-3}-3q^{-4}+3q^{-5}-3q^{-6}+2q^{-7}-2q^{-8}+q^{-9}$
HOMFLY-PT polynomial (db, data sources)	$z^{2}a^{8}+a^{8}-z^{4}a^{6}-3z^{2}a^{6}-3a^{6}+2z^{2}a^{4}+2a^{4}+z^{2}a^{2}+a^{2}$
Kauffman polynomial (db, data sources)	$z^{6}a^{10}-4z^{4}a^{10}+3z^{2}a^{10}+2z^{7}a^{9}-9z^{5}a^{9}+10z^{3}a^{9}-3za^{9}+z^{8}a^{8}-3z^{6}a^{8}+z^{2}a^{8}+a^{8}+3z^{7}a^{7}-13z^{5}a^{7}+16z^{3}a^{7}-7za^{7}+z^{8}a^{6}-4z^{6}a^{6}+6z^{4}a^{6}-6z^{2}a^{6}+3a^{6}+z^{7}a^{5}-4z^{5}a^{5}+7z^{3}a^{5}-4za^{5}+2z^{4}a^{4}-3z^{2}a^{4}+2a^{4}+z^{3}a^{3}+z^{2}a^{2}-a^{2}$
The A2 invariant	$q^{28}-2q^{20}-q^{18}-q^{16}+q^{12}+q^{10}+2q^{8}+q^{6}+q^{2}$
The G2 invariant	$q^{142}-q^{140}+2q^{138}-3q^{136}+q^{134}-q^{132}-3q^{130}+6q^{128}-6q^{126}+4q^{124}-q^{122}-2q^{120}+5q^{118}-4q^{116}+2q^{114}+3q^{112}-3q^{110}+4q^{108}+q^{106}-3q^{104}+8q^{102}-6q^{100}+3q^{98}+q^{96}-4q^{94}+5q^{92}-6q^{90}+3q^{88}-4q^{86}-q^{82}-5q^{80}+q^{78}-4q^{76}-3q^{70}+q^{66}-4q^{64}+6q^{62}-5q^{60}+2q^{58}+4q^{56}-5q^{54}+7q^{52}-2q^{50}+q^{48}+3q^{46}-2q^{44}+q^{42}+2q^{40}+2q^{36}+q^{34}-q^{32}+2q^{30}-q^{28}+q^{26}+q^{24}-q^{22}+2q^{20}+q^{14}+q^{10}$

Further Quantum Invariants

Further quantum knot invariants for 10_133.

A1 Invariants.

Weight	Invariant
1	$q^{19}-q^{17}-q^{13}+2q^{5}+q$
2	$q^{54}-q^{52}-2q^{50}+2q^{48}+q^{46}-2q^{44}+2q^{40}-q^{36}+q^{34}+q^{32}-2q^{30}+q^{26}-2q^{24}-q^{22}+q^{20}-2q^{16}+q^{14}+2q^{12}+q^{6}+q^{4}+q^{2}$
3	$q^{105}-q^{103}-2q^{101}+3q^{97}+3q^{95}-3q^{93}-4q^{91}+4q^{87}+3q^{85}-2q^{83}-4q^{81}-q^{79}+3q^{77}+4q^{75}-2q^{73}-6q^{71}-2q^{69}+6q^{67}+4q^{65}-5q^{63}-4q^{61}+5q^{59}+5q^{57}-3q^{55}-3q^{53}+3q^{51}+3q^{49}-4q^{47}-3q^{45}+q^{43}+3q^{41}-2q^{37}-5q^{35}+2q^{33}+6q^{31}+q^{29}-8q^{27}-6q^{25}+7q^{23}+7q^{21}-3q^{19}-6q^{17}+q^{15}+5q^{13}+2q^{11}-2q^{9}+q^{5}+2q^{3}$

A2 Invariants.

Weight	Invariant
1,0	$q^{28}-2q^{20}-q^{18}-q^{16}+q^{12}+q^{10}+2q^{8}+q^{6}+q^{2}$
1,1	$q^{76}-2q^{74}+4q^{72}-8q^{70}+11q^{68}-14q^{66}+14q^{64}-12q^{62}+8q^{60}-8q^{56}+14q^{54}-19q^{52}+22q^{50}-20q^{48}+20q^{46}-16q^{44}+12q^{42}-6q^{40}+6q^{36}-10q^{34}+12q^{32}-14q^{30}+6q^{28}-8q^{26}-2q^{24}-2q^{20}+2q^{18}+4q^{16}+2q^{14}+6q^{12}+4q^{8}+q^{4}$
2,0	$q^{72}-q^{68}-q^{66}+q^{62}-q^{60}-q^{58}+q^{52}+q^{50}+2q^{48}+3q^{46}+q^{44}-2q^{40}-q^{38}-2q^{36}-2q^{34}-2q^{32}-q^{30}-q^{28}-q^{26}-2q^{22}+2q^{20}+2q^{18}+2q^{16}+q^{14}+2q^{12}+3q^{10}+q^{8}+q^{4}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{60}-q^{58}-2q^{52}-q^{48}+q^{46}+2q^{44}+3q^{42}+3q^{40}+3q^{38}-3q^{34}-4q^{32}-6q^{30}-4q^{28}-3q^{26}+2q^{22}+2q^{20}+3q^{18}+3q^{16}+q^{14}+2q^{12}+2q^{10}+q^{8}+q^{4}$
1,0,0	$q^{37}+q^{33}-2q^{27}-2q^{25}-2q^{23}-q^{21}+q^{17}+2q^{15}+q^{13}+2q^{11}+q^{9}+q^{7}+q^{3}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{78}+q^{72}-q^{70}-3q^{68}-2q^{66}-2q^{64}-4q^{62}-q^{60}+4q^{58}+7q^{56}+6q^{54}+9q^{52}+9q^{50}+2q^{48}-3q^{46}-5q^{44}-11q^{42}-13q^{40}-9q^{38}-7q^{36}-4q^{34}+5q^{30}+5q^{28}+4q^{26}+5q^{24}+5q^{22}+2q^{20}+2q^{18}+2q^{16}+2q^{14}+2q^{12}+q^{10}+q^{6}$
1,0,0,0	$q^{46}+q^{42}+q^{40}-2q^{34}-2q^{32}-3q^{30}-2q^{28}-q^{26}+q^{22}+2q^{20}+2q^{18}+q^{16}+2q^{14}+q^{12}+q^{10}+q^{8}+q^{4}$

B2 Invariants.

Weight	Invariant
0,1	$q^{60}-q^{58}+2q^{56}-2q^{54}+2q^{52}-2q^{50}+q^{48}-q^{46}+q^{42}-3q^{40}+3q^{38}-4q^{36}+3q^{34}-4q^{32}+2q^{30}-2q^{28}+q^{26}+2q^{20}-q^{18}+3q^{16}-q^{14}+2q^{12}+q^{8}+q^{4}$
1,0	$q^{98}-q^{94}-q^{92}+q^{90}+q^{88}-2q^{86}-2q^{84}+q^{82}+2q^{80}-q^{78}-2q^{76}+3q^{72}+2q^{70}+2q^{64}+2q^{62}+q^{60}-q^{58}-q^{56}-q^{52}-4q^{50}-3q^{48}-q^{46}-2q^{42}-3q^{40}+2q^{36}+q^{34}-q^{32}+q^{30}+2q^{28}+3q^{26}+q^{20}+2q^{18}+q^{16}+q^{14}+q^{6}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{82}-q^{80}+q^{78}-2q^{76}+2q^{74}-3q^{72}-2q^{68}+q^{66}+q^{62}+3q^{60}+2q^{58}+6q^{56}+q^{54}+4q^{52}-3q^{50}+q^{48}-7q^{46}-3q^{44}-8q^{42}-4q^{40}-5q^{38}-q^{36}+q^{32}+4q^{30}+2q^{28}+4q^{26}+q^{24}+4q^{22}+2q^{18}+q^{16}+2q^{14}+q^{12}+q^{10}+q^{6}$

G2 Invariants.

Weight

Invariant

1,0

$q^{142}-q^{140}+2q^{138}-3q^{136}+q^{134}-q^{132}-3q^{130}+6q^{128}-6q^{126}+4q^{124}-q^{122}-2q^{120}+5q^{118}-4q^{116}+2q^{114}+3q^{112}-3q^{110}+4q^{108}+q^{106}-3q^{104}+8q^{102}-6q^{100}+3q^{98}+q^{96}-4q^{94}+5q^{92}-6q^{90}+3q^{88}-4q^{86}-q^{82}-5q^{80}+q^{78}-4q^{76}-3q^{70}+q^{66}-4q^{64}+6q^{62}-5q^{60}+2q^{58}+4q^{56}-5q^{54}+7q^{52}-2q^{50}+q^{48}+3q^{46}-2q^{44}+q^{42}+2q^{40}+2q^{36}+q^{34}-q^{32}+2q^{30}-q^{28}+q^{26}+q^{24}-q^{22}+2q^{20}+q^{14}+q^{10}$

.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["10 133"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

$-t^{2}+5t-7+5t^{-1}-t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$-z^{4}+z^{2}+1$

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 19, -2 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

$q^{-1}-q^{-2}+3q^{-3}-3q^{-4}+3q^{-5}-3q^{-6}+2q^{-7}-2q^{-8}+q^{-9}$

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

$z^{2}a^{8}+a^{8}-z^{4}a^{6}-3z^{2}a^{6}-3a^{6}+2z^{2}a^{4}+2a^{4}+z^{2}a^{2}+a^{2}$

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

$z^{6}a^{10}-4z^{4}a^{10}+3z^{2}a^{10}+2z^{7}a^{9}-9z^{5}a^{9}+10z^{3}a^{9}-3za^{9}+z^{8}a^{8}-3z^{6}a^{8}+z^{2}a^{8}+a^{8}+3z^{7}a^{7}-13z^{5}a^{7}+16z^{3}a^{7}-7za^{7}+z^{8}a^{6}-4z^{6}a^{6}+6z^{4}a^{6}-6z^{2}a^{6}+3a^{6}+z^{7}a^{5}-4z^{5}a^{5}+7z^{3}a^{5}-4za^{5}+2z^{4}a^{4}-3z^{2}a^{4}+2a^{4}+z^{3}a^{3}+z^{2}a^{2}-a^{2}$

Vassiliev invariants

V2 and V3:

(1, 0)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 $4$ $0$ $8$ $-{\frac {34}{3}}$ ${\frac {34}{3}}$ $0$ $32$ $-32$ $-32$ ${\frac {32}{3}}$ $0$ $-{\frac {136}{3}}$ ${\frac {136}{3}}$ $-{\frac {1649}{30}}$ $-{\frac {1022}{15}}$ ${\frac {13022}{45}}$ ${\frac {113}{18}}$ ${\frac {1231}{30}}$

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). 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 10 133. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -8 -7 -6 -5 -4 -3 -2 -1 0 χ -1                 1 1 -3               1 1 0 -5             2     2 -7           1 1     0 -9         2 2       0 -11       1 1         0 -13     1 2           -1 -15   1 1             0 -17   1               -1 -19 1                 1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-3$ $i=-1$ $r=-8$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-3$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-1$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

${\textrm {Include}}({\textrm {ColouredJonesM.mhtml}})$

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[10, 133]]
Out[2]=	10
In[3]:=	PD[Knot[10, 133]]
Out[3]=	PD[X[1, 4, 2, 5], X[3, 8, 4, 9], X[14, 9, 15, 10], X[5, 13, 6, 12], X[13, 7, 14, 6], X[18, 11, 19, 12], X[20, 15, 1, 16], X[16, 19, 17, 20], X[10, 17, 11, 18], X[7, 2, 8, 3]]
In[4]:=	GaussCode[Knot[10, 133]]
Out[4]=	GaussCode[-1, 10, -2, 1, -4, 5, -10, 2, 3, -9, 6, 4, -5, -3, 7, -8, 9, -6, 8, -7]
In[5]:=	BR[Knot[10, 133]]
Out[5]=	BR[4, {-1, -1, -1, -2, 1, 1, -2, -3, 2, -3, -3}]
In[6]:=	alex = Alexander[Knot[10, 133]][t]
Out[6]=	-2 5 2 -7 - t + - + 5 t - t t
In[7]:=	Conway[Knot[10, 133]][z]
Out[7]=	2 4 1 + z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[7, 6], Knot[10, 133]}
In[9]:=	{KnotDet[Knot[10, 133]], KnotSignature[Knot[10, 133]]}
Out[9]=	{19, -2}
In[10]:=	J=Jones[Knot[10, 133]][q]
Out[10]=	-9 2 2 3 3 3 3 -2 1 q - -- + -- - -- + -- - -- + -- - q + - 8 7 6 5 4 3 q q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[10, 133], Knot[11, NonAlternating, 27]}
In[12]:=	A2Invariant[Knot[10, 133]][q]
Out[12]=	-28 2 -18 -16 -12 -10 2 -6 -2 q - --- - q - q + q + q + -- + q + q 20 8 q q
In[13]:=	Kauffman[Knot[10, 133]][a, z]
Out[13]=	2 4 6 8 5 7 9 2 2 4 2 -a + 2 a + 3 a + a - 4 a z - 7 a z - 3 a z + a z - 3 a z - 6 2 8 2 10 2 3 3 5 3 7 3 9 3 6 a z + a z + 3 a z + a z + 7 a z + 16 a z + 10 a z + 4 4 6 4 10 4 5 5 7 5 9 5 2 a z + 6 a z - 4 a z - 4 a z - 13 a z - 9 a z - 6 6 8 6 10 6 5 7 7 7 9 7 6 8 8 8 4 a z - 3 a z + a z + a z + 3 a z + 2 a z + a z + a z
In[14]:=	{Vassiliev[2][Knot[10, 133]], Vassiliev[3][Knot[10, 133]]}
Out[14]=	{0, 0}
In[15]:=	Kh[Knot[10, 133]][q, t]
Out[15]=	-3 1 1 1 1 1 1 2 q + - + ------ + ------ + ------ + ------ + ------ + ------ + q 19 8 17 7 15 7 15 6 13 6 13 5 q t q t q t q t q t q t 1 1 2 2 1 1 2 1 ------ + ------ + ----- + ----- + ----- + ----- + ----- + ---- 11 5 11 4 9 4 9 3 7 3 7 2 5 2 3 q t q t q t q t q t q t q t q t