10 146

10_145

10_147

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10 146 Further Notes and Views

Knot presentations

Planar diagram presentation	X4251 X5,18,6,19 X8394 X2,9,3,10 X11,17,12,16 X7,12,8,13 X15,6,16,7 X17,11,18,10 X13,1,14,20 X19,15,20,14
Gauss code	1, -4, 3, -1, -2, 7, -6, -3, 4, 8, -5, 6, -9, 10, -7, 5, -8, 2, -10, 9
Dowker-Thistlethwaite code	4 8 -18 -12 2 -16 -20 -6 -10 -14
Conway Notation	[22,21,21-]

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 [-6][-4] Hyperbolic Volume 10.561 A-Polynomial See Data:10 146/A-polynomial

[edit Notes for 10 146's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $2$ Rasmussen s-Invariant 0

[edit Notes for 10 146's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$2t^{2}-8t+13-8t^{-1}+2t^{-2}$
Conway polynomial	$2z^{4}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 33, 0 }
Jones polynomial	$-q^{3}+3q^{2}-4q+6-6q^{-1}+5q^{-2}-4q^{-3}+3q^{-4}-q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$-z^{2}a^{4}+z^{4}a^{2}+z^{2}a^{2}+z^{4}+z^{2}+1-z^{2}a^{-2}$
Kauffman polynomial (db, data sources)	$a^{2}z^{8}+z^{8}+3a^{3}z^{7}+4az^{7}+z^{7}a^{-1}+3a^{4}z^{6}+a^{2}z^{6}-2z^{6}+a^{5}z^{5}-8a^{3}z^{5}-11az^{5}-2z^{5}a^{-1}-8a^{4}z^{4}-6a^{2}z^{4}+3z^{4}a^{-2}+5z^{4}-2a^{5}z^{3}+5a^{3}z^{3}+12az^{3}+6z^{3}a^{-1}+z^{3}a^{-3}+3a^{4}z^{2}+3a^{2}z^{2}-3z^{2}a^{-2}-3z^{2}-a^{3}z-3az-3za^{-1}-za^{-3}+1$
The A2 invariant	$-q^{16}+q^{14}+q^{12}-q^{10}+q^{8}-q^{6}+q^{2}+2q^{-2}-q^{-4}+q^{-6}+q^{-8}-q^{-10}$
The G2 invariant	$q^{80}-2q^{78}+4q^{76}-7q^{74}+5q^{72}-2q^{70}-6q^{68}+16q^{66}-19q^{64}+20q^{62}-13q^{60}-4q^{58}+19q^{56}-29q^{54}+29q^{52}-14q^{50}-4q^{48}+20q^{46}-22q^{44}+16q^{42}-q^{40}-18q^{38}+23q^{36}-21q^{34}+7q^{32}+13q^{30}-31q^{28}+37q^{26}-24q^{24}+10q^{22}+6q^{20}-27q^{18}+34q^{16}-31q^{14}+22q^{12}-3q^{10}-15q^{8}+28q^{6}-23q^{4}+12q^{2}+4-18q^{-2}+20q^{-4}-13q^{-6}-q^{-8}+22q^{-10}-29q^{-12}+27q^{-14}-11q^{-16}-8q^{-18}+22q^{-20}-27q^{-22}+19q^{-24}-8q^{-26}+q^{-28}+8q^{-30}-12q^{-32}+8q^{-34}-3q^{-36}+2q^{-38}-2q^{-42}-q^{-44}+q^{-48}-q^{-50}+q^{-52}$

Further Quantum Invariants

Further quantum knot invariants for 10_146.

A1 Invariants.

Weight	Invariant
1	$-q^{11}+2q^{9}-q^{7}+q^{5}-q^{3}+2q^{-1}-q^{-3}+2q^{-5}-q^{-7}$
2	$q^{32}-2q^{30}-2q^{28}+6q^{26}-7q^{22}+4q^{20}+4q^{18}-8q^{16}+7q^{12}-3q^{10}-q^{8}+5q^{6}+q^{4}-5q^{2}-1+7q^{-2}-5q^{-4}-5q^{-6}+9q^{-8}-5q^{-12}+4q^{-14}+q^{-16}-2q^{-18}$
3	$-q^{63}+2q^{61}+2q^{59}-3q^{57}-6q^{55}+13q^{51}+5q^{49}-13q^{47}-14q^{45}+8q^{43}+23q^{41}-28q^{37}-13q^{35}+25q^{33}+26q^{31}-18q^{29}-32q^{27}+11q^{25}+33q^{23}-2q^{21}-31q^{19}-4q^{17}+25q^{15}+7q^{13}-19q^{11}-11q^{9}+15q^{7}+16q^{5}-4q^{3}-23q-3q^{-1}+27q^{-3}+14q^{-5}-26q^{-7}-26q^{-9}+22q^{-11}+32q^{-13}-12q^{-15}-32q^{-17}+q^{-19}+26q^{-21}+8q^{-23}-17q^{-25}-8q^{-27}+6q^{-29}+8q^{-31}-q^{-33}-4q^{-35}-q^{-37}+q^{-41}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{16}+q^{14}+q^{12}-q^{10}+q^{8}-q^{6}+q^{2}+2q^{-2}-q^{-4}+q^{-6}+q^{-8}-q^{-10}$
1,1	$q^{44}-4q^{42}+10q^{40}-22q^{38}+38q^{36}-54q^{34}+72q^{32}-86q^{30}+85q^{28}-70q^{26}+42q^{24}-4q^{22}-41q^{20}+86q^{18}-122q^{16}+146q^{14}-153q^{12}+146q^{10}-126q^{8}+96q^{6}-52q^{4}+12q^{2}+34-62q^{-2}+80q^{-4}-92q^{-6}+80q^{-8}-64q^{-10}+43q^{-12}-22q^{-14}+14q^{-16}-2q^{-24}-2q^{-26}+q^{-28}$
2,0	$q^{42}-q^{40}-2q^{38}+3q^{34}+3q^{32}-4q^{30}-2q^{28}+2q^{26}+2q^{24}-3q^{22}-4q^{20}+3q^{18}+2q^{16}-q^{14}+4q^{10}+q^{8}+q^{6}+2q^{4}-2q^{2}-1+q^{-4}-5q^{-6}-2q^{-8}+6q^{-10}+3q^{-12}-3q^{-14}+4q^{-18}+2q^{-20}-2q^{-22}-2q^{-24}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{34}-2q^{32}+2q^{28}-4q^{26}+4q^{24}+2q^{22}-5q^{20}+4q^{18}+q^{16}-5q^{14}+q^{12}+2q^{10}-2q^{8}+q^{4}+3q^{2}-q^{-2}+7q^{-4}-3q^{-6}-2q^{-8}+6q^{-10}-3q^{-12}-3q^{-14}+3q^{-16}-q^{-18}-q^{-20}+q^{-22}$
1,0,0	$-q^{21}+q^{19}+q^{15}-q^{13}+q^{11}-q^{9}+q^{3}+q+2q^{-3}-q^{-5}+q^{-7}+q^{-11}-q^{-13}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{44}-q^{42}-2q^{40}+2q^{38}+q^{36}-3q^{34}+5q^{30}-q^{28}-5q^{26}+2q^{24}+6q^{22}-3q^{20}-4q^{18}+6q^{16}-6q^{12}+2q^{8}-4q^{6}-q^{4}+7q^{2}+3-q^{-2}+5q^{-4}+8q^{-6}-3q^{-8}-3q^{-10}+3q^{-12}-5q^{-16}-2q^{-18}+2q^{-20}+q^{-22}-q^{-24}+q^{-28}$
1,0,0,0	$-q^{26}+q^{24}+q^{18}-q^{16}+q^{14}-q^{12}+q^{4}+q^{2}+1+2q^{-4}-q^{-6}+q^{-8}+q^{-14}-q^{-16}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{34}+2q^{32}-4q^{30}+6q^{28}-6q^{26}+6q^{24}-6q^{22}+5q^{20}-2q^{18}-q^{16}+5q^{14}-7q^{12}+10q^{10}-12q^{8}+12q^{6}-11q^{4}+9q^{2}-6+3q^{-2}+q^{-4}-3q^{-6}+6q^{-8}-6q^{-10}+7q^{-12}-5q^{-14}+5q^{-16}-3q^{-18}+q^{-20}-q^{-22}$
1,0	$q^{56}-2q^{52}-2q^{50}+2q^{48}+4q^{46}-2q^{44}-5q^{42}+7q^{38}+4q^{36}-6q^{34}-6q^{32}+3q^{30}+6q^{28}-6q^{24}-2q^{22}+4q^{20}+3q^{18}-3q^{16}-3q^{14}+3q^{12}+4q^{10}-q^{8}-5q^{6}+6q^{2}+2-5q^{-2}-3q^{-4}+6q^{-6}+5q^{-8}-3q^{-10}-6q^{-12}+2q^{-14}+7q^{-16}+2q^{-18}-5q^{-20}-4q^{-22}+q^{-24}+4q^{-26}-2q^{-30}-q^{-32}+q^{-36}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{46}-2q^{44}+2q^{42}-4q^{40}+5q^{38}-5q^{36}+5q^{34}-5q^{32}+6q^{30}-3q^{28}+2q^{26}-q^{22}+2q^{20}-6q^{18}+6q^{16}-8q^{14}+8q^{12}-10q^{10}+9q^{8}-7q^{6}+10q^{4}-5q^{2}+6-q^{-2}+3q^{-4}+3q^{-6}-3q^{-8}+3q^{-10}-5q^{-12}+6q^{-14}-5q^{-16}+3q^{-18}-5q^{-20}+4q^{-22}-2q^{-24}+q^{-26}-q^{-28}+q^{-30}$

G2 Invariants.

Weight

Invariant

1,0

$q^{80}-2q^{78}+4q^{76}-7q^{74}+5q^{72}-2q^{70}-6q^{68}+16q^{66}-19q^{64}+20q^{62}-13q^{60}-4q^{58}+19q^{56}-29q^{54}+29q^{52}-14q^{50}-4q^{48}+20q^{46}-22q^{44}+16q^{42}-q^{40}-18q^{38}+23q^{36}-21q^{34}+7q^{32}+13q^{30}-31q^{28}+37q^{26}-24q^{24}+10q^{22}+6q^{20}-27q^{18}+34q^{16}-31q^{14}+22q^{12}-3q^{10}-15q^{8}+28q^{6}-23q^{4}+12q^{2}+4-18q^{-2}+20q^{-4}-13q^{-6}-q^{-8}+22q^{-10}-29q^{-12}+27q^{-14}-11q^{-16}-8q^{-18}+22q^{-20}-27q^{-22}+19q^{-24}-8q^{-26}+q^{-28}+8q^{-30}-12q^{-32}+8q^{-34}-3q^{-36}+2q^{-38}-2q^{-42}-q^{-44}+q^{-48}-q^{-50}+q^{-52}$

.

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["10 146"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$2t^{2}-8t+13-8t^{-1}+2t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$2z^{4}+1$

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]=

{ 33, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$a^{2}z^{8}+z^{8}+3a^{3}z^{7}+4az^{7}+z^{7}a^{-1}+3a^{4}z^{6}+a^{2}z^{6}-2z^{6}+a^{5}z^{5}-8a^{3}z^{5}-11az^{5}-2z^{5}a^{-1}-8a^{4}z^{4}-6a^{2}z^{4}+3z^{4}a^{-2}+5z^{4}-2a^{5}z^{3}+5a^{3}z^{3}+12az^{3}+6z^{3}a^{-1}+z^{3}a^{-3}+3a^{4}z^{2}+3a^{2}z^{2}-3z^{2}a^{-2}-3z^{2}-a^{3}z-3az-3za^{-1}-za^{-3}+1$

Vassiliev invariants

V2 and V3:

(0, 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 $0$ $0$ $0$ $-16$ $-16$ $0$ $0$ $32$ $-32$ $0$ $0$ $0$ $0$ $-8$ $-{\frac {32}{3}}$ ${\frac {64}{3}}$ $-{\frac {56}{3}}$ $-8$

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=$0 is the signature of 10 146. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -5 -4 -3 -2 -1 0 1 2 3 χ 7                 1 -1 5               2   2 3             2 1   -1 1           4 2     2 -1         3 3       0 -3       2 3         -1 -5     2 3           1 -7   1 2             -1 -9   2               2 -11 1                 -1

Integral Khovanov Homology (db, data source)

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

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, 146]]
Out[2]=	10
In[3]:=	PD[Knot[10, 146]]
Out[3]=	PD[X[4, 2, 5, 1], X[5, 18, 6, 19], X[8, 3, 9, 4], X[2, 9, 3, 10], X[11, 17, 12, 16], X[7, 12, 8, 13], X[15, 6, 16, 7], X[17, 11, 18, 10], X[13, 1, 14, 20], X[19, 15, 20, 14]]
In[4]:=	GaussCode[Knot[10, 146]]
Out[4]=	GaussCode[1, -4, 3, -1, -2, 7, -6, -3, 4, 8, -5, 6, -9, 10, -7, 5, -8, 2, -10, 9]
In[5]:=	BR[Knot[10, 146]]
Out[5]=	BR[4, {-1, -1, 2, -1, 2, 1, -3, 2, -1, 2, -3}]
In[6]:=	alex = Alexander[Knot[10, 146]][t]
Out[6]=	2 8 2 13 + -- - - - 8 t + 2 t 2 t t
In[7]:=	Conway[Knot[10, 146]][z]
Out[7]=	4 1 + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 146], Knot[11, NonAlternating, 18], Knot[11, NonAlternating, 62]}
In[9]:=	{KnotDet[Knot[10, 146]], KnotSignature[Knot[10, 146]]}
Out[9]=	{33, 0}
In[10]:=	J=Jones[Knot[10, 146]][q]
Out[10]=	-5 3 4 5 6 2 3 6 - q + -- - -- + -- - - - 4 q + 3 q - q 4 3 2 q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[10, 146]}
In[12]:=	A2Invariant[Knot[10, 146]][q]
Out[12]=	-16 -14 -12 -10 -8 -6 -2 2 4 6 8 10 -q + q + q - q + q - q + q + 2 q - q + q + q - q
In[13]:=	Kauffman[Knot[10, 146]][a, z]
Out[13]=	2 3 z 3 z 3 2 3 z 2 2 4 2 z 1 - -- - --- - 3 a z - a z - 3 z - ---- + 3 a z + 3 a z + -- + 3 a 2 3 a a a 3 4 6 z 3 3 3 5 3 4 3 z 2 4 ---- + 12 a z + 5 a z - 2 a z + 5 z + ---- - 6 a z - a 2 a 5 4 4 2 z 5 3 5 5 5 6 2 6 4 6 8 a z - ---- - 11 a z - 8 a z + a z - 2 z + a z + 3 a z + a 7 z 7 3 7 8 2 8 -- + 4 a z + 3 a z + z + a z a
In[14]:=	{Vassiliev[2][Knot[10, 146]], Vassiliev[3][Knot[10, 146]]}
Out[14]=	{0, 0}
In[15]:=	Kh[Knot[10, 146]][q, t]
Out[15]=	3 1 2 1 2 2 3 2 - + 4 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 q t q t q t q t q t q t q t 3 3 3 3 2 5 2 7 3 ---- + --- + 2 q t + 2 q t + q t + 2 q t + q t 3 q t q t